text:
Comparative law and justice/Barbados

Part of the Comparative law and justice Wikiversity Project

Lluangsouphom 20:29, 23 September 2010 (UTC)

Basic Information
Demographics

Total Population: 284,589

Nationality: Barbadian(s) or Bajan (colloquial)

Age structure:
 * 0-14 years: 19.2%
 * male: 27,383
 * female: 27,352

 * 15-64 years: 71.3%
 * male 99,829
 * female 103,049

 * 65 years and over: 9.5%
 * male 10,464
 * female 16,512

Median Age: 36.2 years
 * male: 35.1 years
 * female: 37.2 years

Birth rate: 12.55 births/1,000 population

Death rate: 8.41 deaths/1,000 population

Geography Barbados is located in the Caribbean, island in the North Atlantic Ocean, northeast of Venezuela.

World region/continent: Central America and the Caribbean Total land mass: 430 sq km (half the size of New York City) Terrain: relatively flat, it rises gently to central highland region Climate: tropical; rainy season (July to December); cool season (January to June)
 * overall annual temperature: 75°F to 82°F

Natural hazards: infrequent hurricanes; Neighboring islands and bodies of water: Key mountains: Mount Hillaby Key rivers: Long Pond River in St Andrew, Joe's River in St Joseph and Bruce Vale River in St Andrew.
 * The neighboring countries include: Venezuela and Miami
 * The neighboring bodies of water are: Caribbean Sea and the North Atlantic Ocean

Major Cities:

Religion

→ There is no official religion in Barbados.
 * Protestant 63.4% (Anglican 28.3%, Pentecostal 18.7%, Methodist 5.1%, other 11.3%)
 * Roman Catholic 4.2%
 * other Christian 7%
 * other 4.8%
 * none or unspecified 20.6%

Ethnic Groups

 * black 93%
 * white 3.2%
 * mixed 2.6%
 * East Indian 1%
 * other 0.2%

Barbados' population was dominated by descendants of African slaves, but by the 1980s, a distribution of ethnic groups was typical of the Eastern Caribbean; 90% of the population was black, 5% mulatto, and 5% white. Descendants of the white labor class emigrated from Britain.

Language

The languages spoken in Barbados include English and Bajan (also known as Barbadian Creole English. The offical language is English.

Brief History
The island of Barbados, oftentimes known as "Little England," is located in the Caribbean. Through the centuries, Barbados has remained the most British of the Caribbean Islands. Barbados was first inhabited by the British in 1627 and it wasn't until the gradual introduction of political and social reforms in the 1940s and 1950s that Barbados gained complete independence from the UK in 1966. The island was first inhabited by male settlers and slaves captured en route. In 1639, the House of Assembly was formed, which was the only representative legislature in the Caribbean to remain in existence for more than three centuries.

As a basis for the Constitution of 1966, Barbadians used a statement on individual rights and privileges from the 1652 Charter of Barbados. For almost 300 years, Barbados remained in the hands of a small white minority. It was not until World War I that reform finally came, however, as a result of ideas brought back by Clennell Wilsden Wickham of Barbados, Andrew Arthur Cipriani of Trinidad, and others who had served in the British forces abroad. Wickham returned home in 1919 fired by enthusiasm to make Barbados a more democratic place. The Democratic League succeeded in electing a few representatives to the House of Assembly between 1924 and 1932, but it is chiefly remembered for inspiring O'Neale's nephew, Errol Barrow, to found the Democratic Labour Party (DLP).

"Spontaneous rioting erupted throughout the Commonwealth Caribbean in the late 1930s as the region felt the effects of the worldwide depression. In Barbados, fourteen people were killed and forty- seven wounded in protests in 1937. The rioting pushed Grantley Adams to found the Barbados Labour Party (BLP) in 1938. (The BLP was known briefly as the Barbados Progressive League.) Adams, a lawyer who had won the Barbados Scholarship to Oxford in 1918, became the most important figure in preindependence politics. He quickly rose to prominence through his testimony before the British Moyne Commission. Adams argued that the main cause of the riots was economic distress. Elected to the House of Assembly in 1940, Adams became president general of the Barbados Workers Union (BWU) on its formation in 1941."

"Under Barbadian governor Sir Grattan Bushe, the constitution was changed to effect a semiministerial form of government, and the franchise was progressively liberalized. During the 1942 House of Assembly session, Adams led a fight for reforms that broadened the franchise by reducing the cost of qualification, increased direct taxation, established a workmen's compensation program, and protected union leaders from liability in trade disputes." Adams became leader in 1946. Independence was gained and Errol Barrow became its first prime minister.

Economic Development, Health, and Education
Economic Development

GDP: $3.595 billion GDP per capita: $17,700 Average annual income: Industries: tourism, sugar, light manufacturing, component assembly for export Imports: Exports:
 * $1.586 billion
 * commodities: consumer goods, machinery, foodstuffs, construction materials, chemicals, fuel, electrical components
 * partners: Trinidad and Tobago 28.52%, US 27.96%, Colombia 7.13%, China 4.76%, UK 4.39%
 * $385 million
 * commodities: manufactures, sugar and molasses, rum, other foods and beverages, chemicals, electrical components
 * partners: Trinidad and Tobago 17.48%, Jamaica 15.63%, US 8.93%, Saint Lucia 8.13%, UK 5.36%, Saint Vincent and the Grenadines 5.04%, Antigua and Barbuda 4.12%

Health

Infant mortality rate:12.29 deaths/1,000 live births
 * male: 13.89 deaths/1,000 live births
 * female: 10.67 deaths/1,000 live births

Life expectancy: 73.94 years
 * male: 71.65 years
 * female: 76.26 years

Education

Literacy rate: 99.7%
 * male: 99.7%
 * female: 99.7%

→The average education attainment stands so far as everyone age 15 and over has ever attended school. Barbados has had a longstanding emphasis on education, and is known for having one of the oldest and most advanced education systems in the Eastern Caribbean.

Governance

Barbados is a common law country. "At independence in November 1966, Barbados formally adopted the Westminster parliamentary system of government, with a governor general representing the British monarch. Rights and privileges accorded to the governor in 1652 by Britain formed the basis for the Constitution of 1966, which provides for a bicameral parliamentary system headed by a prime minister and cabinet." Executive authority in Barbados relies upon the decisions of the governor general, the prime minister, and a cabinet of at least five ministers. "During the first twenty years of Barbadian history, all of its governments remained in power until the five-year limit."

Barbados' Parliament consists of the British monarch, which is represented by the governor general, the Senate, and the House of Assembly. The monarch appoints the governor general, who serves to the monarch. Duties of the governor general include: appointing judges, commissioners, and senators and for voting in the Senate if there is a tie, but the governor general must act in accordance with the advice of the cabinet or one of its ministers. The governor general presides at all meetings of the Privy Council for Barbados. The Privy Council is an appointed body whose duties include the right to review punishments and grant pardons.

The prime minister holds the most power in the executive and within the cabinet. The prime minister holds the power to choose the cabinet ministers and may also dismiss them. The cabinet, which is responsible to Parliament, is the principal instrument of policy and is charged with direction and control of the government. However, the personality, style, and popularity of the prime minister largely determines the direction of government.

History of Governance in Barbados

The DLP and BLP, which are both centrist social democratic parties and Barbados' two major parties, hold power in organizing the government since independence in 1966. Power is evenly divided between the two parties, and there is little difference between the parties, although it is said that the DLP is a little more conservative than the BLP. "Both parties strongly support private enterprise, but both have undertaken large public works as a necessity in a country where unemployment ranges between 15 and 20 percent."

Constitution

Barbados has a Constitution. The Constitution consists of a House of Assembly of 24 members or as many as Parliament chooses. However, the number was increased to 27 before the 1981 elections. Members of the House of Assembly must be over 21 years old and are elected by universal suffrage. The leader of the majority in the House of Assembly is appointed prime minister by the governor general. The minority leader becomes leader of the opposition. The term of office is five years, but elections may be called at any time by the ruling party, and an election must be called in case of a vote of no confidence.

The Constitution contains various public service commissions which oversees government acts. The commissions include: the Judicial and Legal Service Commission, the Public Service Commission, and the Police Commission. After deliberation with the Prime minister and leader of the opposition, commissioners are appointed by the governor general.

Laws

A bill becomes a law after it has passed both the Senate and the House of Assembly and has been signed by the governor general. Bills may be introduced in either The Senate or House of Assembly. Money bills are an exception, they must be introduced in the elected House of Assembly.

Elections
The Senate is an exclusively appointed body. Senators must be citizens of Barbados and over the age of twenty-one. The Senate elects its own president and deputy president.

Barbados' judiciary includes the Supreme Court, which consists of the High Court and the Court of Appeal. The governor general appoints the chief justice after consultation with the prime minister and the leader of the opposition. The other judges who make up the court are appointed by the Judicial and Legal Service Commission. Judges serve until the age of sixty-five. Appeals from decisions made by the High Court may be made to the three-judge Court of Appeal. The highest appeal is to the Judicial Committee of the Privy Council in London.

Voting for General Elections

Barbadian citizens, aged 18 or older are required to vote in a general election on polling day. Elections are a secret ballot and are anonymous. Voting takes place on election day from 6:00 a.m. to 6:00 p.m. Polling stations are in public buildings such as schools and churches.

On election day, Barbadians vote for candidates running in each of the 30 single-seat constituencies throughout Barbados. A party needs to win 16 constituencies to command a majority in the House of Assembly, which allows it to choose a Prime Minister, formally appointed by the Governor General. Once selected, the prime minister begins the task of forming a government.

Voting is by secret ballot. The only people allowed in the polling station are: the officer who is in charge, the polling clerks, the duty police officers, the candidates, their election agents and polling agents and the voters. Voting takes place in a booth, which is monitored to maintain secrecy. Ballots are placed in the ballot box, which is then sealed and delivered to the counting station.

Judicial Review
The judicial review for Barbados refers to the high court or special senate of the high court. The Supreme Court provides the uniformity for jurisdiction. The decisions for a rule take effect only if all parties are in agreement.

Courts and Criminal Law
Barbados' criminal law system follows ordinary court proceedings.

The judiciary system in Barbados includes the Supreme Court, which consists of the High Court and the Court of Appeal. The chief justice is appointed by the governor general after consultation with the prime minister and the leader of the opposition. Appeals from decisions made by the High Court may be made to the three-judge Court of Appeal. The highest appeal is to the Judicial Committee of the Privy Council in London.

The key actors in courts are the judge and the jury. Their responsibilities include hearing and decision making during court cases.

Role of judges and selection The judges make up the Supreme Court. Judges are appointed by the Judicial and Legal Service Commission. Judges serve until the age of 65. In case of vacancies of judges, the governor general then has the authority to appoint acting judges who serve until the appropriate consultations have been made. A judge cannot enter the duties of his office until he has taken an oath of allegiance and a judicial oath.

Role for jury Barbados uses a jury system for their court hearings and cases. 12 jurymen (and women) make up the jury.

Role of lawyers, educated, trained Lawyers in Barbados receive their education from The University of the West Indies at Cave Hill, Mona, St. Augustine, and the University of Guyana. Graduates go on to become leading members of the bar, the bench and the legal academic community.

Lawyers attend school for the undergraduate and postgraduate program. The undergraduate course leads to the LLB degree (Bachelor of Law) and requires full time attendance for three years. The program is divided into three parts: Students must complete each part successfully, and there is limited acceptance into Parts II and III. The academic year is divided into 2 semesters (from September to December, and from January to May). Students must take 5 courses per semester, 3 credits each. They must have 90 credits in order to be awarded with the degree. In the final year, students are allowed to choose their own program of study, including courses and seminars. Students however, are encouraged to pursue their own interests such as interning, independent study under faculty supervision, and other interests concerning lawyer ambition. Students in their final year are given the opportunity to study abroad at the University of Toronto (Canada)/UWI Exchange Program for one semester.
 * Part I is offered at the University of The West Indies, St Augustine, Mona, The College of the Bahamas and Cave Hill Campuses.
 * Parts II and III are only offered at the Cave Hill campus. The undergraduate program is only offered to the students from countries which contribute to the Faculty of the Schools.

Graduate Studies Program: Specialized Programs offerred by the Faculty of Law:

The attorney in Barbados prepares all court documents and represents individuals in courts. After attaining your law degree, lawyers in barbados decide if they would like to work in the private or public sector. Lawyers in Barbados are trained and educated in schools located in the Bahamas, Jamaica or Trinidad. They do this during the course of their undergraduate and postgraduate work in school. Undergraduates are given a chance to work and intern with faculty members and experiences lawyers during their course work. Postgraduates enroll into programs suiting their needs.
 * The Master of Laws in Corporate and Commercial Law
 * The Advanced Diploma in Corporate and Commercial Law
 * General LL.M.
 * the LL.M. in Public Law
 * the LL.M. in International Law
 * the LL.M. in Judicial Studies
 * the LL.M. in Legislative Drafting

Barbados follows ordinary court proceedings. Defendants are allowed lawyers, the accused does have a chance to speak in court. It is presumed that an accused defendant is innocent until proven guilty.

Organization of courts There is a Supreme Court of Judicature is the high court of court of appeal. Under the constitution, they have unlimited jurisdiction, consisting of a chief justice and three puisne judges, who are appointed by the governor general, who follows under the prime minister. Magistrate courts are part of the civil and criminal jurisdiction. The eight nations--Eight nations—Barbados, Belize, Dominica, Guyana, Jamaica, St. Lucia, St. Vincent and the Grenadines, and Trinidad and Tobago; have met to try and form the Caribbean Court of Justice (CCJ). The judiciary in Barbados is independent and free from any political influences.

"The courts enforce respect for civil rights and assure a number of due process protections in criminal proceedings including a right of detainees to be brought before a judge within 72 hours of arrest."

Punishment
Punishment of crime in Barbados is worth of the crime the offender has committed. The smallest of crimes such as theft of small items, those committing can be sent to prison and must pay a fine of however much the cost of the item stolen was. The Barbadian police force have a "zero tolerance" policy. The policeman in Barbados are allowed to use excessive force. Policeman however, see this is a way of defense, while others argue it is police brutality.

Barbados is still for the death penalty, but the last execution was in 1983. The country feels as though it is mandatory to keep the death penalty, 92 percent of the citizens want to maintain it.

There are prostitution laws, and although there has been talk of child trafficking, it has not been proven or documented that Barbados partakes in child trafficking.

Since 2005, Barbados has been working building a new prison, which will house 1200 inmates, including women and men. The prisoners will be allowed access to health services, recreational facilities and skills training. Brutality in prison can however result to trial, conviction and a sentence in court. It has been very common that officers in prison assault inmates, and like anyone else can face criminal charges. Barbados prison officers hold this act true to their heart, it has been reported that during the past ten years, there have been deaths in prisons, resulting from asthma attacks, heart attacks, and AIDS.

The Nation's prison rate: the prison population total, pertaining to numbers of people in penal institutions including pre-trial detainees: 992 out of the nation's population of 270,000. The prison population rate is 367 per 100,000 of nation's population.

Corporal punishment was not outlawed in schools, but it was subject to guidelines, namely, that it must not be used excessively or inhumanely, he said. The Government and people of Barbados do not view corporal punishment as torture, or inhumane or degrading in itself. But, if corporal punishment was used improperly, it could result in inhumane or degrading treatment. It is permitted in schools only under strict regulation and only by a principal or senior teacher. As so flogging (beating, whipping) children in reform schools, the law has allowed it only to be used for difficult or unreasonable children in certain types of circumstances. Many members of the government argue to rid of corporal punishment in schools, whereas it could lead to torture, death, etc., yet many others feel it is necessary to keep in order to teach disciplinary rules to children.

Law Enforcement
Divisions of Law Enforcement

"The Royal Barbados Police Force is charged with maintaining public order. The small volunteer Barbados Defence Force (BDF) is responsible for national security and can be employed to maintain public order in times of crisis, emergency, or other specific need.  Police committed some human rights abuses."

Law Enforcement Officers The Royal Barbados Police Force is composed of 200 special constables, and 1,250 police officers. Entry in to the Royal Barbados Police Force is through a selection process. Individuals, however, need to meet certain prerequisites, which are:

 * Age:
 * Police Constables: Not less than 19 and not more than 30 years of age. Persons over the age of 30 years may be enlisted as Police Constables in the following cases:
 * -Special Constables who have served not less than three (3) years in the force.
 * -Persons who have served not less than two years in a police force.

 * Special Constables:Not less than 19 and not more than 40 years of age.

 * Academic Qualifications: At least three (3) C.X.C. subjects (General) and grade I, II or III or Three (3) subjects at G.C.E. (Ordinary) for Police Constables. These qualifications are not required of an applicant for the post of Special Constable.

 * Citizenship: Citizen of a Commonwealth Country.

 * Character: Good character and conduct, high integrity.

 * Health: Physically and mentally fit.

Upon receiving the application at the Training Office, the selection process occurs in 8 stages:

Stage 1→ Initial Interview: An interview with the Recruiting Officer and another member of the staff. This interview will be based on the information in your application. Stage 2→ Written Entrance Examination: This examination is comprised of: English, Arithmetic, an Essay and General Knowledge. The exam tests: grammar, written communication skills, numeric skills and knowledge of current affairs. Stage 3→ Intelligence Test: The test is administered by the Ministry of Education. It is designed to test: perceptiveness and ability to make timely and logical decisions. Stage 4→ Blood Tests and Chest X-Rays: The tests are to ensure that there are no existing abnormalities likely to affect training and subsequent service as a police officer. Stage 5→ Physical Fitness Evaluation: This is comprised of a series of exercises and a timed run over a measured distance. It is designed to test: agility, strength and stamina. Stage 6→ Medical Examination: A complete medical examination by the Force's Medical Officer to ensure that applicants are in good medical health. This examination includes testing of: eyes, heart, lungs and blood pressure. Stage 7→ Psychiatric Evaluation: Applicants will be assessed by a trained psychologist to determine whether individuals are psychologically fit for police work. Stage 8→ Formal Panel Interview: This panel comprises members of the Selection Board, headed by an Assistant Commissioner of Police. The file will be examined in detail and, in addition to other questions, applicants may be queried about information contained therein. (The Board determines suitability to be enlisted as a Police Officer).

The selection process may take up to as long as 4 months. Upon passing the selection stage, Police Constables train for 22 weeks at the Regional police Training Centre, while Special Constables train for 7 weeks. Applicants can expect to have investigations run on their background, and can be eliminated at any stage of the process if they do not meet certain criteria. If applicants fail the Entrance Examination of the fitness evaluation, they may reapply after a year, however, individuals can not reapply if they fail any of the other components of the selection process.

Barbados ranks 7.4 out of 10 on the Corruptions Perception Index.

The Royal Barbados Police force focuses on slowing down drug trafficking and enforcing drug laws. The Barbados policeman work with the United States Coast Guard to combat drug trafficking, while also making use of British-trained drug sniffing dogs. Careful watch of drug trafficking is very important in Barbados because the Caribbean region is a major transit for drugs coming from South America to Europe and to the United States. The Barbados Defense Force is responsible for national security.

Role of Military --there is no military structure for the country of Barbados.

Crime Rates and Public Opinion
Crime Rates

Crime in Barbados is moderate compared to other industrialized countries. The typical types of crime include: murder, forcible rape, robbery, aggravated assault, burglary, larceny, and motor vehicle theft "Barbados has below average rates of vehicle theft, theft from vehicles, vehicle vandalism, robbery and assault." The country has lower average crime rates for sex offenses, personal theft, corruption and consumer fraud. The only crime where Barbados has a relatively high rate is burglary, this also includes attempts. "Barbados has a much higher rate of reporting crime to the police than most other countries." Of all countries, Barbados has the highest reporting rates of robbery, threats, sex offenses and assaults. The source of this data is from surveys, police reports and victimization surveys.

Public Opinion

The Barbados Crime Survey (2002) shows that a majority of Barbadians: "over-estimate the increase in general crime, over-estimate the increase in homicide rates, believe a great deal more crime is violent than it is; and underestimate the severity of the courts." 42% of the people agree that the courts are too lenient, Barbadians feel as though the sentencing for burglary is far too lenient, 62% feel the burglars should be imprisoned. They felt a major way to prevent crime in the country starts in the home, almost a third thought better schooling was the answer and 37% thought it was to be tackled by reducing poverty and increasing employment." 50% of the country agrees that the police forces are helpful in controlling crime, and an average number of people felt safe in their homes.

Family Law
Marriage Requirements for obtaining a marriage license:
 * Both parties must be present at the Ministry of Home Affairs
 * The following documents are required:
 * 1) Valid Passport and the original or certified copies of the applicants’ birth certificates, and identification cards;
 * 2) If either party was married previously and widowed, an original or certified copy of the marriage certificate and death certificate in respect of the deceased spouse;
 * 3) If either party was divorced, an original Decree Absolute or Final Judgement. The Decree Nisi is not accepted.

 * Additional requirements:
 * 1) Persons who have been grated Barbadian citizenship must present original document
 * 2) Non-nationals that have been granted a work permit must present a passport stating that the work permit has been granted
 * 3) No marriage license will be issues to a person whose visitor's stay has expired

Marriage of minors: Persons that are 16 years and younger cannot be legally married in Barbados. Persons that are 18 years and older can be married an do not require parental consent. Minors must pay Fees: Arrangements must be made with a Marriage Officer, before completing an application form for a marriage license. A letter from the Marriage Officer is required at the time the application if filled out.
 * Bds$200.00 cash and a $25.00 stamp – If neither party is a citizen or resident of Barbados.
 * Bds$45.00 cash and a $10.00 stamp – If either party is a citizen of Barbados.

There are no laws in Barbados that discriminate women. Barbados encourages participation of women in the government and public affairs. Women are allowed equal access to education, employment, and health care. Politically, within the past couple of years, women have quickly become more involved. There have been a high number of women candidates. Of the 18 cabinet members, five of them were women, including ones at the higher levels of the cabinet.

Adoption Barbados is not a part of the Hague Convention on Protection of Children and Co-operation in Respect of Intercountry Adoption. Only citizens of countries that have diplomatic relations to barbados an adopt Barbadian children. The U.S has diplomatic relations with Barbados. Who can Adopt? Children to be adopted must meet the requirements and definition of an orphan.
 * Residency requirements: The adoption of a Barbadian child can take place in Barbados or in the applicants’ country of residence. Applicants wishing to adopt a child in Barbados will be required to live on the island for at least 18 months. If the applicants want the adoption to take place in the country which they are currently living in, the applicants are required to come to Barbados for at least a few weeks in order to receive the child into their care. They must also attend the High Court hearing for the license to adopt the child and take the child out of Barbados.
 * Age requirements: If prospective parents are already related to the child they want to adopt, they must be at least 18 years old. But, if there are no relations, one of the parents must be 25 years old and at least 18 years older than the child.
 * Marriage requirements: Both married and single people may adopt a child from Barbados

The Process of adopting a child from Barbados:
 * 1) Choose an Adoption Service Provider
 * 2) Apply to be Found Eligible to Adopt
 * 3) Be Matched with a Child
 * 4) Adopt the Child in Barbados
 * 5) Apply for the Child to be Found Eligible for Adoption
 * 6) Bring Your Child Home

Divorce Requirements to get a divorce in Barbados:
 * The couple must keep the one case before the same judge for all of the court proceedings
 * The couple must attend counseling during the process of the divorce
 * Decisions of property and ownership of shared items can be deferred until the divorce is finalized
 * Decisions of financial issues and property can still continue even if one of the parties become deceased during the process
 * If support payments are involved, either of the parties must continue to make payments

In the case of inheritance, a widow, or family of those who have deceased is entitled to inherit a portion of the spouses' land.

It is arguable and mostly agreed that children should be treated better in Barbados. Corporal punishment still exists in schools, and many citizens have a problem with that. On the other hand, elders are treated with the utmost respect. There are programs and policies that protect the rights of the elders.

Human Rights
The fundamental rights protected by the legal system are:
 * Fundamental rights and freedom of the individual
 * Protection of right to life
 * Protection of right to personal liberty.
 * Protection from slavery and forced labour
 * Protection from inhuman treatment.
 * Protection from deprivation of property
 * Protection against arbitrary search or entry
 * Provisions to secure protection of law
 * Protection of freedom of conscience
 * Protection of freedom of expression
 * Protection of freedom of assembly and association
 * Protection of freedom of movement
 * Protection from discrimination on grounds of race, etc.
 * Enforcement of protective provisions
 * Time of emergency
 * Saving of existing law
 * Interpretation

Every citizen in Barbados has the right to equality and freedom of speech. Barbados is very passionate and proud of providing human rights and equality to all their citizens. Barbados promotes equality for all.

"Persons with disabilities, are “near and dear” to both the Government and people of Barbados. The country had done much to develop and enhance policies and programmes to protect people with disabilities. The Government was also fully supportive of the new international Convention on the issue, and it had passed legislation to promote the equalization of opportunities."

title: Comparative law and justice/Barbados


text:
Comparative law and justice/Kenya

Part of the Comparative law and justice Wikiversity Project

Basic Information
Kenya is located Eastern Africa, bordering the Indian Ocean, between Somalia and Tanzania. The geographic coordinates are 1 00 N, 38 00 E. The total area of Kenya is 580,367 sq km, the land total is 569,140 sq km and the water total is 11,227 sq km. The country Kenya is slightly more than twice size of Nevada. Kenya’s land boundaries are a total 3,477 km. The border countries are Ethiopia 861 km, Somalia 682 km, Sudan 232 km, Tanzania 769 km, and Uganda 933 km. Kenya’s coastline stands at 536 km, the  Kenya’s climate varies from tropical along coast to arid in the interior. The elevation extremes lowest point is the Indian Ocean at 0 m the highest point stands at 5,199 m that is Mount Kenya. The natural resources of Kenya are limestone, soda ash, salt, gemstones, fluorspar, zinc, diatomite, gypsum, wildlife, and hydropower. The ethnic groups that reside in Kenya are “Kikuyu 22%, Luhya 14%, Luo 13%, Kalenjin 12%, Kamba 11%, Kisii 6%, Meru 6%, other African 15%, non-African (Asian, European, and Arab) 1%” "Protestant 45%, Roman Catholic 33%, Muslim 10%, indigenous beliefs 10%, other 2%" The two official languages spoken in Kenya are english and kiswahili, there are also numerous other languages spoken.

Brief History
President Jomo Kenyatta was who led Kenya from 1963 until 1978. His time as president was cut short due to his unfortunate death due to old age. Eventually a new president was brought in by the name of Danial Toroitich. A lot of Kenya’s country success is credited to Danial Toroitich.

Economic Development, Health, and Education
The GDP the (purchasing power Parity) was recorded at 62.48 billion in with a growth rate at 2.6% the year 2009. The GDP-per capita was $1,600 in the year 2009, with 200 other countries similar. Kenya’s exports were at $4.459 billion in the year 2009, slightly lower than 2008 when it was at $5.04 billion. The exports that are disturbed from Kenya are tea, horticultural products, coffee, petroleum products, fish, and cement. The export partners of Kenya are” UK 11.31%, Netherlands 9.81%, Uganda 9.07%, Tanzania 8.83%, US 5.93%, Pakistan 5.63%” as of 2009. Kenya’s imports come from ” India 11.67%, China 10.58%, UAE 9.32%, South Africa 8.36%,Saudi Arabia 6.53%, US 6.25%, Japan 5.1%” in 2009. Imports Kenya receives are transportation equipment, petroleum products, iron, steel, plastic, and motor vehicles. In  2009 an estimated $ 9.715 billion was spent on imports. The Infant mortality rate stands at a median of 53.49 deaths out of 1000 live births. The male and female rates are quite similar, but there is a slight difference with males at 56.28 and females at 50.63 out of 1000 births each. It shows that 44 countries have similar information. The Median of the life expectancy in Kenya stands at 58.82 years out of the total population. Female have a slight higher life expectancy rate at 59.32 over males who are at 58.33 years.

Governance, Elections, and Judicial Review
Kenya has three forms of government executive,legislative, and judicial. The chief of state for the executive branch is Mwai Kibaki and his vice president is stephen Kalonzo Musyoka. The president is not in control of any of the government business, the prime minster mr.Raila odinga, is in charge of that. In Kenya the cabinet is selected by the president and is lead by the prime minister. Kenya's national assembly 224 seats and 210 members, the members are elected and are expected to serve a five year term. The president is the person who selects 12 members but they must still be elected by their parties.Court of Appeal (chief justice is appointed by the president); High Court Each president that is elected doe's a five year term and has an option to do a second term. When running for president you must get the most overall votes but must win 25% of the votes in five out of Kenya's seven provinces. the last election was held in december 2007 and the next one will be december 2012. Kenya has an republic government ad was given their independence december 12, 1963. They receives independence from the British of east Africa.

Courts and Criminal Law
The Kenya court system has 5 different systems.the 5 courts is the court of Appeal High Court that handles moving courts to one region to another, the Resident Magistrate's Court, the District Magistrates Courts and the traditional courts. The high court holds a total of 12 judges. The chief of Justice, which is another way of calling the head judge is appointed by the President of Kenya. The president is also incharge of appointing the the other 11 judges

Punishment
There are many Penalties for criminal in Kenya just like any other common countries. A criminal can receive minor punishment such as community service, probation, and fines. The punishment can also be as much as a determinate prison sentences, life in prison without parole, and the death penalty. One of the most common crimes to be punished for is murder. The maximum penalty is death and its done by being hung, but you can also be sentenced to life in prison depending on circumstances.the prison conditions are not so bad, it shows that they can recieve time off for good behavior and time added for bad behavior. When you are in prison you must be registered in some kind of education program. The dangerous offenders are exempt form the education programs. Inmates with good behavior are allow to go on weekend leave, play sports, and have group therapy in some prisons. To be prosecuted as an adult you must be over the age of 17 to be tried as an adult. There are some ways a minor can be tried as an adult with a crime such as murder. There were no reports of favoritism ethnic group, gender, and religion when it comes to legal punishment in Kenya.

Law Enforcement and Legal Personnel
In Kenya the police the police are trained by going to a Kiganjo Police college. In fact some of Kenya's police recruits are sent over for special training in Europe and America. some of the training the recruits have done are similar to what is done by our FBI agency. The education requirements can be a high school diploma but some still hold college degrees. When a suspect is brought in for questioning some police force people to admit to committing a crime. These methods are Usually used when the police don't catch the suspect in the act of the crime. The Kenya Armed forces include Kenya’s army, navy, and air force. The military obligation is a minimum of 9 years and the navy is only 7 years. In order to apply to the Kenya army you must a citizen of Kenya and present a national identity card. A national identity card you can only receive at the age of 18, u also needs a school leaving certificate. You are able to join younger than 18 be having a parents’ consent.

Crime Rates and Public Opinion
In the country of Kenya crime is common due to the poverty being at 50% and the unemployment being at 40%. The typical crimes that are committed are murder, force able rape, theft, and drug offenses. The reported homicide rate was recorded 3.5% of 100,000 of the population. The crime rate for force able rape was recorded at 1.3% of 100,000 and a highest rated crime was recorded at 72.9% out of 100,000. I would say it is so high due to the poverty rate being so high. Drug offense are report to also be very common.

Family Law
The common marriage style in Kenya was polygamy. Kenyan society has quickly transited into a monogamous one. Polygamy is having more then one spouse, it was never uncommon for a a man to have multiple wives. In order for a man to marry a woman, he must get approval of the woman father. usually in order to marry a mans daughter you are subject to a fee. The 1st wife is the most expensive one. the inheritance is passed from father to son but women can get the inheritance. they must go through a legal and cultural process to be granted the inheritance.

Human Rights and Social Inequality
The economic, social and cultural rights say that you have the rights to food, housing, water, healthcare, sanitation, social security, Liberty, privacy, religion, belief,education, and political rights. The people of Kenya are protected by any kind of discrimination to persons with disabilities, older members of society, youth and children.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Comparative law and justice/Kenya


text:
User:Egm6321.f10.team5.steinberg/hw2

Given
A first order nonlinear ordinary differential equation (N1-ODE) may be represented as follows:

$$ F(x,y,y') = M(x,y) + N(x,y)f(y') = 0 \! $$

Find
Find $$ f(y') \! $$ such that there is no analytical solution to $$ f(y') = - \frac{M}{N} \! $$ (i.e. such N1-ODE cannot be exact).

Solution
The first condition of exactness is defined as follows.

For $$ F(x,y,y') = 0 \! $$ to be exact, it must be in the form $$ F(x,y,y') = M(x,y) + N(x,y)y' = 0 \! $$.

From this condition, it is understood that if $$ f(y') \not= y' \!$$, then $$ f(y') \! $$ must be solved explicitly for $$ y' \!$$ in order to force the equation into the form shown in the first condition definition above.

For the present problem, we are interested in finding a function $$ f(y') \! $$ that cannot be solved explicitly. To state this another way, we desire $$ f(y') \!$$ such that $$ f^{-1}(y') \!$$ cannot be determined analytically.

Let $$ f(y') = y'(3+y') \!$$, for example.

$$ F(x,y,y') = M(x,y) + N(x,y)f(y') = M(x,y) + N(x,y)y'(3+y') = 0 \! $$

$$ y'(3+y') = - \frac{M}{N} \! $$

Clearly, this expression cannot be solved explicitly for $$ y' \! $$. Furthermore, the N1-ODE is not exact because this particular $$ f(y') \!$$ cannot satisfy the first condition of exactness.

Given
$$ a(x) = sin(x^3) \! $$

$$ b(x) = cos(x) \! $$

$$ c(y) = exp(2y) \! $$

Find
1. Using $$ \bar b(x)c(y)y'+a(x) \bar c(y) = 0 \! $$, find a N1-ODE that is either exact or can be made exact.

2. Find $$ \phi(x,y) = k \! $$.

Solution
$$ \bar b(x) = \int^x b(s) ds = \int^x cos(s) ds = sin(x) \! $$

$$ \bar c(y) = \int^y c(s) ds = \int^y exp(2s) ds = \frac{1}{2} exp(2y) \! $$

Substituting these along with $$a(x) \!$$ and $$c(y) \!$$ into the given equation yields the following N1-ODE:

$$ \frac{1}{2} sin(x^3)exp(2y) + sin(x)exp(2y)y' = 0 \! $$

In this form, the N1-ODE is not exact since the second condition of exactness is not satisfied i.e. $$ \frac{\partial M}{\partial y} \not= \frac{\partial N}{\partial x} $$.

However, we may employ the Euler integrating factor method to force the equation to be exact.

$$ hM + hNy' = 0, \bar M_y = \bar N_x \! $$

$$ \bar M_y = h_y M + h M_y \! $$

$$ \bar N_x = h_x N + h N_x \! $$ $$ h_x N - h_y M + h(N_x - M_y) = 0 \! $$

$$ M = \frac{1}{2} sin(x^3) exp(2y) \! $$

$$ N = sin(x) exp(2y) \! $$

$$ M_y = sin(x^3) exp(2y) \! $$

$$ N_x = cos(x) exp(2y) \! $$

Let $$ h_x = 0 \!$$

$$ h(y) = exp(\int \frac{1}{M} (N_x - M_y) dy) $$

$$ = exp \Bigg ( \int \frac{2}{sin(x^3)exp(2y)} exp(2y) [ cos(x) - sin(x^3) ] dy \Bigg) $$

$$ = exp\Bigg (y \bigg [\frac{cos(x)}{sin(x^3)}-1 \bigg] \Bigg) $$

??????

title: User:Egm6321.f10.team5.steinberg/hw2


text:
Northern Arizona University/Environmental Ethics/Journals/Angela's Journal

Journal Entry 1

Leopold describes the different layers in the land pyramid as being so intricately intertwined as to appear random. However, the stability with which this system operates indicates that it is, in fact, highly organized. I think this stability is why nature can be so therapeutic. Among the disorder in our lives caused by human variables, nature, even through its vicissitudes, is constant. Its densely interwoven parts are too vast and too mutable to fully comprehend. From our perspective, nature can therefore appear to be an unconquerable foe. This makes it the ideal place to find solace from the disorder in our lives.

However, nature is not unconquerable. However small our species may be in the biotic community, the land pyramid is still subjected to our influences. Agriculture and grazing strips the soil of its nutrients. Hunting eliminates species. Countless other influences surely play a role, but these processes would better be described by an ecologist than by a psychology major. Nevertheless, it is our responsibility as conscious beings within the pyramid to preserve this structure as best we can. Leopold states that, "obligations have no meaning without conscience." Therefore, we must first recognize our place within this elaborate system and create ways to lessen our negative impact upon it.

I think it is important to stipulate that human nature will not allow ecological mindedness to supersede more basic needs, such as the nee for food or the need for safety. This could explain the inaction of the famers during the Soil Conservation. After being promised free technical service and specialized machinery in exchange for writing roes for land use, the farmers had not written any rules even after a decade. However, this law was passed in 1937 during the Great Depression. A decade later, World War II had only been over for two years. I think ecological mindedness is among the next steps in our growing domain of concern as a society. However, first it is necessary that more basic needs are met.

Journal Entry 2

Early this morning I drove out to Mormon Lake to visit family. Although I missed the sunrise by a couple hours, I caught some great views of autumn's imminent presence on the variegated landscape. For me, reading Sand County Almanac has been as much intellectually stimulating as it has been a guilt trip. As I drove, I wondered how much plant-life had been driven out for the sake of the road and my convenience. I wondered how many species of animals were disturbed or communities destroyed. I thought about how survival is a constant struggle in the natural world, whereas in American cities and towns, existence is a right we grant ourselves. As the dominant species, I believe this right was collectively earned. However, as thinking beings, I believe we are also charged with the responsibility of preserving nature. Surely, we all consider ourselves above any single bunny. This is easily justified by our longer life spans, greater capacity for feeling and connectedness, and greater potential for having a positive impact on others. However, when comparing ourselves, as one member of one species, to the complexity of nature, our own significance is hard to perceive.

The communities we've built for ourselves can be complemented by nature but never substituted for it. In "July," Leopold notes the amount of wild plant species to be found in the suburb and campus as compared to the backward farm. The farm outweighed the suburb and campus by 226 to 120. Leopold referred to this count as the "total visual diet." He seems to suggest that nature is better prepared to meet our fundamental needs than are communities. Pointing to my cell phone and laptop, I might argue that nature does not possess all the tools I require, but perhaps the same can be said about our cities. There must be tools and knowledge in nature which cannot be duplicated. It seems that it would be in our best interest to preserve nature, and therein, the mysteries it holds. However, these "mysteries" may be difficult to state in economic terms.

No matter how much thinking power we are endowed, I think humans will always be ruled primarily by self-interest. It is difficult to adapt to new values and even more difficult to persuade others to do so. Therefore, I think concern for preserving the biotic community will only increase as its benefit to us becomes more evident. As our understanding of the scientific and our recognition of the aesthetic grows, these values could become integrated into our culture.

Journal Entry 3

“I have read many definitions of what is a conservationist and written not a few myself, but I suspect that the best one is written not with a pen, but with an axe” (p. 68). In the November chapter, Leopold seems to be making a distinction between thought and action. As well-intentioned as an individual might be, having (or proclaiming to have) a sense of ecological responsibility does not necessarily benefit the greater community. Only when our actions are consistent with these beliefs, can we consider ourselves ecologically responsible. To Leopold, a conservationist is someone who takes careful consideration in deciding what to chop and does so “humbly aware that with each stroke he is writing his signature on the face of his land” (p. 68). The best person to understand, appreciate, and preserve nature is the one immersed within it.

At one point, Leopold seems to take a god-like stance, in relation to the environment. With a shovel and an axe, he says that whoever owns land is endowed with the power to give and take life. However, along with his own influence, he recognizes that every other living thing has the same power to its own extent. Death and rebirth is a delicate and necessary balance to continue renewing life; each species’ power to give and take life is a way of maintaining this balance.

Leopold goes on to suggest that nature is at his mercy just as much as he is at its mercy. Through his passive observation of a thriving community of plants and animals, Leopold seems to be acknowledging his humble status as a single entity within it. For him, this is evidence that he is better suited to simply enjoy his surroundings rather than attempting to control it or derive selfish favors from it.

Journal Entry 4

I think a major appeal of John Muir’s writings are his seeming lack of concern over his own safety. He recalls venturing into the Florida marshlands for several days with just some bread in his pocket. In Yosemite, he describes cataclysmic storms and flooding as he gazes on passively and unprotected. These events are consistent with his assertion that people can live more fully by immersing themselves in nature’s struggle for survival rather than insulating themselves from threat.

In Darwinian theory, nature involves chance: random genetic variation, the survival of some, and the death of others. For Muir, this is a key part of nature’s inherent sublimity and beauty. Nature is a force infinitely greater than our selves, and it submits to nothing. Yet, its power extends from the interrelatedness of its components. This interrelatedness emerges from giving and taking life. Therefore, danger and risk is inherent in nature. Humans represent only a single component and are subject to this danger and risk. No matter how much we try to avoid it through existing in modern society, we are still subject to the whims of others and susceptible to the smallest disease-causing microbes.

Muir suggests that such danger and risk in life should be embraced. This attitude may make a person’s life significantly shorter than remaining sheltered in society, but this small concern is heavily outweighed by the fulfillment that a person stands to experience by embracing nature and seeing beauty that manifests through uncertainty.

Journal Entry 5

Throughout Economy, Thoreau seems to be making the argument that our needs can be divided up into two categories: those of a physical nature and those of a spiritual nature. Like Socrates, he is suggests that our spiritual needs are of greater value than our physical needs. Therefore, we must preserve our virtues, at the cost of our comforts and even at the cost of our life if necessary. Furthermore, I think he is saying that it is each person's responsibility to develop himself/herself intellectually, emotionally, and spiritually.

Thoreau's model of moral living brought to mind the hierarchy of needs created by a famous psychologist, Abraham Maslow. At the bottom, are our physiological needs (food, water), followed by our safety needs (security of body, employment, morality), followed by needs for love and belonging (friendship, family, sexual intimacy), followed by esteem needs (self-esteem, confidence, achievement, respect), and finally our need to self-actualize (being moral, creative, spontaneous, accepting). Maslow argued that the next step in the pyramid could only be met when all the lower needs were met. Furthermore, he proposed that only a very limited number of people actually reach self-actualization.

My purpose in comparing the two models is to show aspects of our lives that Thoreau's model does not seem to directly address. Nevertheless, I think Thoreau did consider such aspects and could categorize these needs into his model. The needs in question are safety needs, love and belonging, and esteem needs. By seeing how Thoreau lived his life, we can understand how much value he placed on each of these. He seemed to set the base of the pyramid by establishing his basic needs before beginning his sojourn. During these two years approximately he had little or no contact with civilization. Thus precluding the achievement of needs for either safety or love and belonging. Thus, it seems that to Thoreau these are hindrances that prevent one from meeting their spiritual needs. Esteem needs, on the other hand, are intrapersonal. Therefore, how well these needs are met depend on the expectations the individual sets for himself/herself. Thoreau seemed to have clear direction for his experiment, and through his readings it seems he was confident in himself. Although I do not think his needs included recognition from others, I think his esteem needs were met.

Journal Entry 6

“To be awake is to be alive. I have never yet met a man who was quite awake. How could I have looked him in the face?” Thoreau argues that every morning millions wake up ready for manual labor, but of those, only a small portion are awake for intellectual exertion, and of these, an even smaller portion possesses poetic consciousness. Thoreau praises morning as a time of re-invigoration and innocence in which we make ourselves as innocent and simple as nature.

As he scorns the use of “mechanical aids” to induce wakefulness, I immediately think of my morning coffee. Sometimes I rely on this to give me the mental stimulation I need to complete my responsibilities for the day. However, a narrow-minded view of daily tasks threatens our ability to appreciate the world in a broader sense. Thoreau speaks of expectation and elevating his life to a higher standard. The dawn is symbolic of both of these as we are faced with the infinite possibilities of the day and the beauty it has to offer to the conscious observer.

In his effort to embrace both possibility and beauty, Thoreau retreats to the woods. He attempts to make his life fulfilling even in its most minute details. He limits his affairs to only the most essential matters: shelter, food, hygiene. This frees his conscious energy to continually renew himself spiritually. In our daily lives, I think a majority of our energy is expended on routine tasks. Personally, I find it easy to see these tasks as important in and of themselves, when, in fact, they only exist to support the things that are truly important. Our job, our studies, even our professional ambitions only define a part of who we are. However, by casting off titles and responsibilities, as during the innocence and simplicity of morning, we can awaken and be our fundamental selves.

Journal Entry 7

In the chapter, “The Bean-Field,” I was surprised to learn that Thoreau refers to his experiment as somewhat of a failure. He describes his toil in planting the seeds of sincerity, truth, simplicity, faith, innocence, and the like in himself. However, he says that the seeds, if indeed they were seeds of these virtues, were either wormeaten or had lost their vitality. Planting these seeds requires regular hoeing of the weeds, or our appetitive desires, to keep them in check. A healthy amount of these is, nevertheless, necessary for our virtues to thrive. It is this side of us, the appetitive side, that produces the vitality with which we pursue virtue. When we neglect our animal instincts we begin to let customs guide us rather than acting of our own account. Fate, in a way, begins to pull us in so that we are not so much directing our behavior as we are puppets performing according to our past experience and others’ expectations of us.

Thoreau notes that he regards the rational side and the animal side as equally important. He makes a conscious attempt to embrace this animal side, yet it seems to contradict the purpose behind our appetitive instincts if we must consciously convince ourselves to attend to them. Just as the beauty of a concept, such as life, eludes us when explained in scientific terms, so perhaps does nature when too much thought is applied to it.

In my own life, this is a concept I have struggled to reconcile. Does rational thought come at the expense of appreciating the aesthetic quality of life, or are the two somehow complementary? Once we consciously interpret a feeling, does it cease to be a feeling? Thoreau argues that our rational side only grows with the proper amount of appetitive desire. Yet, sometimes the balance seems impossible to achieve, or the two seem so disparate that they need separate fields in which to grow and thrive.

Journal Entry 8

Much of the first and second chapters of Emerson’s essays seem to explain his conception of the universal mind. It is something which is timeless and inherent within us. It is a finite number of simple truths from which is derived an infinite number of expressions.

These qualities seem to mirror Thoreau’s definition of virtue. Both cross all generations, essentially rendering it timeless. Thoreau attributes the driving force of virtue as coming from our appetitive side while Emerson, similarly, posits that it comes from our intuition, or the natural force within us. “Every man discriminates between the voluntary acts of his mind, and his involuntary perceptions, and knows that to his involuntary perceptions a perfect faith is due” (p. 39). Here, he recognizes the rational side, but seems to be focusing on the mysteriousness of the more primitive influences of our nature.

However, at least in the assigned readings, Emerson seems to address the relationship between the individual and society more in depth. He clearly encourages non-conformity, and, as such, places little value on social norms and laws that do not reflect inherent values, such as education or justice. On p. 10, he states, “What the former age has epitomized into a formula or rule for manipular convenience, it will lose all the good of verifying for itself, by means of the wall of that rule.” Emerson seems to be implying that humans have a tendency to accept rules and customs unquestioningly. In order to exercise our characters to the fullest extent, we must rid ourselves of imitation and, for better or worse, embrace our unique identities.

Journal Entry 9

Emerson’s essay on friendship is at first poetic and optimistic enough to restore anyone’s faith in people. He says that the sweetness of life comes from our feelings of love and good will toward others. We are naturally inclined to gravitate toward the presence of others even though we may not realize it. This inherent affection gives us continual hope of finding people who genuinely hear us and understand us.

In the later part of the chapter, Emerson seems to almost contradict the importance he places on friendship by discussing the importance of solitude. He embraces solitude, yet is careful not to overindulge the pleasure he finds in the company of others. In this regard, impatience is balanced by bashfulness and apathy so that the relationship can ripen on its own schedule.

Emerson’s argument seems consistent with his views of self-reliance. In the earlier chapter, he suggests that the best way of tapping into the common soul is to focus on developing ourselves. As we learn to tune out distraction, we tune in to our intuition and the natural force guiding us. Therein, we give expression to universal truths, and strengthen our bond to all people. Likewise, with friendships we must first focus on developing ourselves. Emerson states that to have friends you must first be a friend. He embraces solitude because it allows him to be a better friend by keeping his own ideas and visions fresh. Although this argument is perhaps selectively applicable to introverts, it seems logical that we must first allow ideas to thrive within ourselves before we are able to share them, or share ourselves, with others.

Journal Entry 10

In Emerson’s later chapters, he seems to outline more in depth how growth occurs by coming to know what he refers to as “the eternal one.” Truth, beauty, and goodness are the qualities of the universe, which enable our understanding of the universe to grow outwardly in all directions by concentric circles. As we discover universal truths, we better understand the nature of the common soul. However, even by examining ourselves, this soul is never to be fully understood. Emerson compares this to a stream whose source is hidden. Although the source (the common soul) may never be found, the stream (the soul as expressed through people) helps us to better understand its nature and live our own lives accordingly.

Furthermore, Emerson argues that each element in nature represents a part of a greater whole. However, we can only see the world piece by piece because we employ constructs such as space and time, but for universal truths, space and time are meaningless. Our disjointed perspective seems to take shape not only in the physical forms of the world, but in events in our lives. We segment our lives into work, school, friends, etc. and deal with each part on its own terms. However, by having a presence of mind to recognize beauty and truth, we can see that each moment is perfect. Emerson says the spirit plays with time and can “crowd eternity into an hour, or stretch an hour to eternity” (p. 158). Instead of measuring our experience along the linear scale of time, we can better connect our experiences by placing the significance in each individual moment. Although the context of the moments will vary, each aims to better grasp truth and beauty in the universe. Thus, each moment is an opportunity to expand our circle of knowledge and to ascribe meaning to our lives.


text:
User:Egm6321.f10.team5.elliott

hw1

hw2 problem7

hw2 problem8

title: User:Egm6321.f10.team5.elliott


text:
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title: User talk:Eno~enwikiversity


text:
User:Egm6321.f10.team5.elliott/hw2

Heading text
Homework 2, Problem 8

Find: $$\left(\alpha \right)$$ show that k1 is not necessary.

If we integrate left side of(6) pg 10-3(lecture) it will reveal how k1 cancels during integration.

 * $$ \int_{}^{x} \frac{ {h}_{x}}{h}dx= \int_{}^{s}n \left(s \right)ds$$

 * $$ \left[ \ln \left(h \right)+ {k}_{1}\right]- \left[ \ln \left(h \left(0 \right) \right)+ {k}_{1} \right]= \int_{}^{s}n \left(s \right)ds$$

 * {| style="width:100%" border="0" align="left"

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * $$\ln \left(h \right)+ \ln \left(h \left(0 \right) \right)+\left[{k}_{1}- {k}_{1} \right]= \int_{}^{s}n \left(s \right)ds$$
 * $$\ln \left(h \right)+ \ln \left(h \left(0 \right) \right)+\left[{k}_{1}- {k}_{1} \right]= \int_{}^{s}n \left(s \right)ds$$

k1 term cancels on left side
 * }
 * }
 * }

$$\left( \beta \right)$$ show that equation (6) pg 10-3(lecture) agrees with King pg 512.

The linear, 1st order, ODE in King is written in the form:

 * $$ \frac{dy}{dx}+P \left(x \right)y=Q \left(x \right)$$

The structure of the solution in King is written as:

$$y= {y}_{h}+ {y}_{p}$$

where the particular solution is in the form:

$$ {y}_{p}= \exp \left\{- \int_{}^{x}P \left(x \right)dt \right\} \int_{}^{x}Q \left(s \right) \exp \left\{ \int_{}^{s}P \left(t \right)dt \right\}ds$$

However, equation 6 pg10-3 (lecture) is a particular solution written in the form:

 * $${y}_{p} \left(x \right)= \frac{1}{h \left(x \right)} \int_{0}^{x}h \left(s \right)b \left(s \right)ds$$

Where,from (1) and (5) pg 10-2 (lecture) we get:

 * $$h \left(x\right)= \exp \left[ \int_{}^{x}n \left(s \right)ds\right]$$

 * $$b \left(x\right)= \frac{R \left(x \right)}{P \left(x \right)}=Q \left(x \right)$$

 * $$n \left(x\right)=P \left(x \right)$$

Finally,plugging h(x),b(x), n(x) into the particular solution form given by King we get:

 * {| style="width:100%" border="0" align="left"

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * $$ {y}_{p}= \exp \left\{- \int_{}^{x}h \left(s \right)ds\right\} \int_{}^{x}n \left(s \right) \exp \left\{ \int_{}^{x}b \left(s \right)ds \right\}ds$$
 * $$ {y}_{p}= \exp \left\{- \int_{}^{x}h \left(s \right)ds\right\} \int_{}^{x}n \left(s \right) \exp \left\{ \int_{}^{x}b \left(s \right)ds \right\}ds$$

 * }
 * }
 * }

'''$$ \left( \gamma \right)$$ solve the following equation:

$$\frac{dy}{dx}+ {a}_{o}y=0$$

$$ {a}_{o}=x $$

solve by separation of variables:

$$ \int_{}^{y} \frac{dy}{y}+ \int_{}^{x}xdx=0$$

$$ \ln \left(y \right)+ \frac{ {x}^{2}}{2}=C$$

$$\ln \left(y \right)=-\frac{ {x}^{2}}{2}+C$$

 * {| style="width:100%" border="0" align="left"

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * $$y= \exp \left[- \frac{ {x}^{2}}{2}+C \right]= {y}_{h}$$
 * $$y= \exp \left[- \frac{ {x}^{2}}{2}+C \right]= {y}_{h}$$

 * }
 * }
 * }

title: User:Egm6321.f10.team5.elliott/hw2


text:
User:Egm6321.f10.team2.patterson/Homework 1

Given
The horizontal force acting on a wheel/ magnet travelling along a flexible guideway can be modeled by the following equation:

$$\begin{align} & {{c}_{0}}\left( {{y}^{1}},t \right)=-{{F}^{1}}\left\{ 1-\overline{R}u_{ss}^{2}\left( {{y}^{1}},t \right) \right\}-{{F}^{2}}u_{s}^{2}\left( {{y}^{1}},t \right)-{}^{T}\!\!\diagup\!\!{}_{R}\;+M\left\{ \left\{ 1-\overline{R}u_{ss}^{2}\left( {{y}^{1}},t \right) \right\}\left\{ u_{tt}^{1}\left( {{y}^{1}},t \right)-\overline{R}u_{stt}^{2}\left( {{y}^{1}},t \right) \right\}+u_{s}^{2}\left( {{y}^{1}},t \right)u_{tt}^{2}\left( {{y}^{1}},t \right) \right\} \\ & \\ \end{align}$$

Find
Perform a dimensional analysis on all of the terms in the given equation and provide an explanation of its physical meaning

$$\begin{align} & {{c}_{0}}\left( {{y}^{1}},t \right)=-{{F}^{1}}\left\{ 1-\overline{R}u_{ss}^{2} \right\}-{{F}^{2}}u_{s}^{2}-{}^{T}\!\!\diagup\!\!{}_{R}\;+M\left\{ \left\{ 1-\overline{R}u_{ss}^{2} \right\}\left\{ u_{tt}^{1}-\overline{R}u_{stt}^{2} \right\}+u_{s}^{2}u_{tt}^{2} \right\} \\ & \left[ {{c}_{0}}\left( {{y}^{1}},t \right) \right]=-\left[ {{F}^{1}} \right]\left\{ \left[ 1 \right]-\left[ \overline{R} \right]\left[ u_{ss}^{2} \right] \right\}-\left[ {{F}^{2}} \right]\left[ u_{s}^{2} \right]-{}^{\left[ T \right]}\!\!\diagup\!\!{}_{\left[ R \right]}\;+\left[ M \right]\left\{ \left\{ \left[ 1 \right]-\left[ \overline{R} \right]\left[ u_{ss}^{2} \right] \right\}\left\{ \left[ u_{tt}^{1} \right]-\left[ \overline{R} \right]\left[ u_{stt}^{2} \right] \right\}+\left[ u_{s}^{2} \right]\left[ u_{tt}^{2} \right] \right\} \\ \end{align}$$

Solution
The dimensional analysis of each term of the given equation is detailed below:

$$\begin{align} & \left[ {{F}^{1}} \right]=\left[ {{F}^{2}} \right]=ML{{T}^{-2}}\text{ } \\ & \left[ \overline{R} \right]=\left[ R \right]=\left[ {{u}^{1}} \right]=\left[ {{u}^{2}} \right]=L \\ & \left[ u_{ss}^{2} \right]={{L}^{-1}} \\ & \left[ u_{s}^{2} \right]=1\text{ } \\ & \left[ T \right]=M{{L}^{2}}{{T}^{-2}}\text{    } \\ & \left[ u_{tt}^{1} \right]=\left[ u_{tt}^{2} \right]=L{{T}^{-2}} \\ & \left[ u_{stt}^{2} \right]={{T}^{-2}} \\ \end{align}$$

Inserting each dimension into the equation yields the following:

$$\begin{align} & \left[ {{c}_{0}}\left( {{y}^{1}},t \right) \right]=-F\left\{ \left[ 1 \right]-L{{L}^{-1}} \right\}-F\cdot 1-{}^{M{{L}^{2}}{{T}^{-2}}}\!\!\diagup\!\!{}_{L}\;+M\left\{ \left\{ \left[ 1 \right]-L{{L}^{-1}} \right\}\left\{ L{{T}^{-2}}-L{{T}^{-2}} \right\}+1\cdot L{{T}^{-2}} \right\} \\ & \left[ {{c}_{0}}\left( {{y}^{1}},t \right) \right]=-F-F-ML{{T}^{-2}}+M\left\{ 1\left\{ L{{T}^{-2}} \right\}+L{{T}^{-2}} \right\}=-F-F-F+F+F \\ & \left[ {{c}_{0}}\left( {{y}^{1}},t \right) \right]=F \\ \end{align}$$

Each term in the given equation has the following physical explanation:
 * The axial force on the wheel, F1, which is influenced by the distance from the guideway centroid and the center of the wheel/magnet and the curvature of the guideway
 * The transverse force on the wheel projected into the axial direction, F2 multiplied by the transverse displacement u2s
 * The axial force generated from the Torque applied to the wheel (not the case for a magnet), $${}^{T}\!\!\diagup\!\!{}_{R}\;$$
 * Mass of the wheel multiplied by it's acceleration.

The sum of each term equals the horizontal force acting on a wheel/ magnet.

Given
Show why $${{c}_{3}}\left( {{Y}^{1}},t \right){{\ddot{Y}}^{1}}$$ is nonlinear with respect to Y1

Solution
A nonlinear ordinary differential equation (ODE) is as an ODE that is not linear. There are few properties of ODEs that distinguish them as linear. First, the dependent variable and all of the derivatives of the dependent variable are of a degree one. Second, nonlinear functions of the dependent variable or its derivatives aren’t allowed.1

In general, linear ODEs must satisfy the following statement:

$$F\left( \alpha u+\beta v \right)=\alpha F\left( u \right)+\beta F\left( v \right)$$

Evaluating the left hand side (LHS) of the equation for $$F\left( {{Y}^{1}},t \right)={{c}_{3}}\left( {{Y}^{1}},t \right){{\ddot{Y}}^{1}}\left( t \right)$$ yields:

$$\begin{align} & F\left( \alpha u+\beta v \right)={{c}_{3}}\left( \alpha u+\beta v,t \right){{{\ddot{Y}}}^{1}}\left( t \right) \\ & \text{                  }=M\left[ 1-\overline{R}u_{ss}^{2}\left( \alpha u+\beta v,t \right) \right]{{{\ddot{Y}}}^{1}}\left( t \right) \\ \end{align}$$

An evaluation of the right hand side (RHS) of the equation yields:

$$\begin{align} & \alpha F\left( u \right)+\beta F\left( v \right)=\alpha {{c}_{3}}\left( u,t \right){{{\ddot{Y}}}^{1}}\left( t \right)+\beta {{c}_{3}}\left( v,t \right){{{\ddot{Y}}}^{1}}\left( t \right) \\ & \text{                        }=\alpha M\left[ 1-\overline{R}u_{ss}^{2}\left( u,t \right) \right]{{{\ddot{Y}}}^{1}}\left( t \right)+\beta M\left[ 1-\overline{R}u_{ss}^{2}\left( v,t \right) \right]{{{\ddot{Y}}}^{1}}\left( t \right) \\ & \text{                        }=M\left[ \alpha \left( 1-\overline{R}u_{ss}^{2}\left( u,t \right) \right)+\beta \left( 1-\overline{R}u_{ss}^{2}\left( v,t \right) \right) \right]{{{\ddot{Y}}}^{1}}\left( t \right) \\ \end{align}$$

It is evident that the RHS does not equal the LHS and the expression is nonlinear because

$$F\left( \alpha u+\beta v \right)\ne \alpha F\left( u \right)+\beta F\left( v \right)$$

1Zill, D., and Cullen, M. Differential Equations with Boundary-Value Problems. Pacific Grove, CA: Brooks/ Cole,


text:
User:Egm6321.f10.team2.patterson/Homework 2

= Problem 7: Ordinary Differential Equations with Varying Coefficients (ODE_VC) =

Given
Show that the constant, k1, is not necessary when the following expression is evaluated:

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$$ \displaystyle
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 * style="width:95%" |

y\left( x \right)=\frac{1}{h\left( x \right)}\int_ – ^{x}{h\left( s \right)b\left( s \right)ds}

$$ (eq 7.1) [1]
 *
 * }

Solution
The function h(x), given in Equation 7.1, is further expressed as:

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$$ \displaystyle
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h\left( x \right)={{e}^{\int_ – ^{x}{{{a}_{0}}\left( s \right)ds}}}

$$ (eq 7.2) [2]
 *
 * }

By converting the integral using the following principle:
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$$ \displaystyle
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\int_ – ^{x}{f\left( s \right)ds}=\int{f\left( x \right)dx}+k $$ (eq 7.3) [2]
 *
 * }

The function h can be expressed as follows:
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$$ \displaystyle
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h\left( x \right)={{e}^{\left( \int{{{a}_{0}}\left( x \right)dx} \right)+k}}={{e}^{k}}{{e}^{\int{{{a}_{0}}\left( x \right)dx}}} $$ (eq 7.4)
 *
 * }

Setting the constant ek to k yields the following:

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$$ \displaystyle
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h\left( x \right)={k}{{e}^{\int{{{a}_{0}}\left( x \right)dx}}} $$ (eq 7.5)
 *
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Plugging Equation 7.4 into both instances of h(x) in Equation 7.1 yields the following:

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$$ \displaystyle
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y\left( x \right)=\frac{1}\int_ – ^{x}{{{k}_{2}}{{e}^{\int{{{a}_{0}}\left( x \right)dx}}}b\left( s \right)ds}=\frac{{e}^{-\int{{{a}_{0}}\left( x \right)dx}}}\int_ – ^{x}{{{e}^{\int{{{a}_{0}}\left( x \right)dx}}}b\left( s \right)ds}

$$ (eq 7.6)
 *
 * }

Let the constant k2=k3k1 which therefore makes k1 unnecessary.

Given
Show that Equation 7.1 agrees with the following:

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$$ \displaystyle
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y\left( x \right)=A{{y}_{h}}\left( x \right)+{{y}_{p}}\left( x \right)

$$ (eq 7.7) [3]
 *
 * }

Solution
Applying the principle in Equation 7.3 directly into Equation 7.1 yields the following:

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$$ \displaystyle
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y\left( x \right)=h{{\left( x \right)}^{-1}}\left[ \int{h\left( x \right)b\left( x \right)dx}+k \right]

$$ (eq 7.8)
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
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y\left( x \right)=kh{{\left( x \right)}^{-1}}+h{{\left( x \right)}^{-1}}\int{h\left( x \right)b\left( x \right)dx}

$$ (eq 7.9)
 *
 * }

It is shown that the two equations agree by letting A=k and making the following substitutions:

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$$ \displaystyle
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{{y}_{h}}\left( x \right)=h{{\left( x \right)}^{-1}}

$$ (eq 7.10)
 *
 * }

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$$ \displaystyle
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{{y}_{p}}\left( x \right)=h{{\left( x \right)}^{-1}}\int{h\left( x \right)b\left( x \right)dx}

$$ (eq 7.11)
 *
 * }

Given
Find yH independently by solving the following equation:

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$$ \displaystyle
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y'+{{a}_{0}}y=0

$$ (eq 7.12)
 *
 * }

Solution
Equation 7.12 can be converted into the following form:

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$$ \displaystyle
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\frac{dx}+{{a}_{0}}{{y}_{H}}=0

$$ (eq 7.13)
 *
 * }

In solving for yH, it follows that

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$$ \displaystyle
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{}^{d{{y}_{H}}}\!\!\diagup\!\!{}_\;=-{{a}_{0}}dx

$$
 *
 * }

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$$ \displaystyle
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\int{{}^{d{{y}_{H}}}\!\!\diagup\!\!{}_\;}=-\int{{{a}_{0}}dx} $$
 *
 * }

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$$ \displaystyle
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\ln \left( {{y}_{H}} \right)=-\int{{{a}_{0}}dx}+k $$
 *
 * }
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$$ \displaystyle
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{{y}_{H}}=K{{e}^{-\int{{{a}_{0}}dx}}}$$
 *

 * }

= Problem 8: Exact Differential Equations =

Given
Find a N1_ODE that is either exact or can be made exact using the following equation:

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$$ \displaystyle
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\bar{b}\left( x \right)c\left( y \right)y'+a\left( x \right)\bar{c}\left( y \right)=0

$$ (eq 8.1)
 *

 * }

With the following values
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$$ \displaystyle
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a\left( x \right)=\sin \left( {{x}^{3}} \right) $$ (eq 8.2)
 *

 * }
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$$ \displaystyle
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b\left( x \right)=\cos \left( x \right) $$ (eq 8.3)
 *

 * }
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$$ \displaystyle
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c\left( y \right)={e}^{2y} $$ (eq 8.4)
 *

 * }

Solution
The functions with superscript bars are integrals of the original functions and thus the following equations are derived:

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$$ \displaystyle
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b\left( x \right)=\cos \left( x \right)\Rightarrow \bar{b}\left( x \right)=\sin \left( x \right) $$ (eq 8.5)
 *

 * }
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$$ \displaystyle
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c\left( x \right)={{e}^{2y}}\Rightarrow \bar{c}\left( x \right)={}^{1}\!\!\diagup\!\!{}_{2}\;{{e}^{2y}} $$ (eq 8.6) Inserting all of the values into Equation 8.1 yields the following:
 *

 * }
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$$ \displaystyle
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\underbrace{\sin \left( x \right)\left( {{e}^{2y}} \right)}_{N\left( x,y \right)}y'+\underbrace{\sin \left( {{x}^{3}} \right)\left( {}^{1}\!\!\diagup\!\!{}_{2}\;{{e}^{2y}} \right)}_{M(x,y)}=0 $$ (eq 8.7) Equation 8.7 meets the first condition of exactness which states that:
 *

 * }
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$$ \displaystyle
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N\left( x,y \right)y'+M\left( x,y \right)=0 $$ (eq 8.8)
 *

 * }

Divide Equation 8.7 by e2y.

Next we can check to see if the equation passes the second condition of exactness in which My must equal Nx
 * {| style="width:100%" border="0"

$$ \displaystyle
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{{M}_{y}}\left( x,y \right)=0 $$ (eq 8.9)
 *

 * }
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$$ \displaystyle
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{{N}_{x}}\left( x,y \right)=\cos \left( x \right) $$ (eq 8.10) From the previous equations it is evident that
 *

 * }
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$$ \displaystyle
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{{M}_{y}}\left( x,y \right)\ne {{N}_{x}}\left( x,y \right) $$ (eq 8.11) Therefore, in order to make the equation exact, Euler's integrating factor, h(x) must be derived as follows:
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle h\left( x \right)={{e}^{-\int{\frac{1}{N}\left( {{N}_{x}}-{{M}_{y}} \right)dx}}} $$ (eq 8.12)4
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle ={{e}^{-\int{\frac{1}{\sin \left( x \right)}\left( \cos \left( x \right)-0 \right)dx}}} $$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
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$$ \displaystyle ={{e}^{-\ln \left( \sin \left( x \right) \right)}} $$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

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$$ \displaystyle h\left( x \right)=\frac{1}{\sin \left( x \right)} $$ (eq 8.13)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

In order to verify that the derived integrating factor is correct it is necessary to multiply Equation 8.8 by h(x) and denote the new values for M and N as follows:
 * {| style="width:100%" border="0"

$$ \displaystyle \underbrace{\left( \frac{1}{\sin \left( x \right)} \right)\left( \sin \left( x \right) \right)}_{\bar{N}=hN\left( x,y \right)}y'+\underbrace{\left( \frac{1}{\sin \left( x \right)} \right)\left( \frac{\sin \left( {{x}^{3}} \right)}{2} \right)}_{\bar{M}=hM\left( x,y \right)}=0 $$ (eq 8.14)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Again we check to see if the second condition of exactness has been met as follows:

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$$ \displaystyle {{\bar{M}}_{y}}\left( x,y \right)=0 $$ (eq 8.15)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$ \displaystyle {{\bar{N}}_{x}}\left( x,y \right)=0 $$ (eq 8.16)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

The N1_ODE has been made exact since

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$$ \displaystyle {{\bar{M}}_{y}}\left( x,y \right)={{\bar{N}}_{x}}\left( x,y \right) $$ (eq 8.17)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Given
Find the first integral of the N1_ODE used in the previous problem so that

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$$ \displaystyle
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\phi \left( x,y \right)=k

$$ (eq 8.18)
 *

 * }

Solution
Considering the following equations:
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$$ \displaystyle
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 * style="width:95%" |

M\left( x,y \right)={{\phi }_{x}}\left( x,y \right) $$ (eq 8.19)5
 *

 * }

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

N\left( x,y \right)={{\phi }_{y}}\left( x,y \right) $$ (eq 8.20)5
 *

 * }

The following equations can be derived:
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$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\int{M\left( x,y \right)dx}+w\left( y \right)=\phi \left( x,y \right) $$ (eq 8.21)6
 *

 * }
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$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\int{N\left( x,y \right)dy}+v\left( x \right)=\phi \left( x,y \right) $$ (eq 8.22)6
 *

 * }

By setting equations 8.21 and 8.22 equal to each other, the functions w(y) and v(x) can be found. From there, equation 8.13 can be solved.
 * {| style="width:100%" border="0"

$$ \displaystyle \int{M}\left( x,y \right)dx+w\left( y \right)=\int{N}\left( x,y \right)dy+v\left( x \right) $$ (eq 8.23)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle \int{\frac{\sin \left( {{x}^{3}} \right)}{2\sin \left( x \right)}dx}+w\left( y \right)=y+v\left( x \right) $$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$ \displaystyle \phi \left( x,y \right)=\int{\frac{\sin \left( {{x}^{3}} \right)}{2\sin \left( x \right)}dx}+y=k $$
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

= References = 1Vu-Quoc,L. EGM 6321.F10, 10-3, Equation (6)

2Vu-Quoc,L. EGM 6321.F10, 11-1:11-2

3King, A.C. et al. Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge: Cambridge University Press

4Vu-Quoc,L. EGM 6321.F10, 10-1, Equation (6)&(7)

5Vu-Quoc,L. EGM 6321.F10, 8-2, Equations (1)&(2)

6Zill, D., and Cullen, M. Differential Equations with Boundary-Value Problems. Pacific Grove, CA: Brooks/ Cole,


text:
User talk:Egm6321.f10.team5.oh/hw2

= Problem #1 =  Verify Non-linear 1st Order ODE(N1-ODE)

Given

 * {| style="width:100%" border="0" align="left"

$$\displaystyle M(x,y) + N(x,y)y' = 0 =: F(y) $$ $$\displaystyle (1-1) $$
 * }
 * }

Find
1) Verify the above equation is in General N1-ODE

2) Give an example for L1-ODE

Solve
== > (1-1) is Ordinary Differential Equation(ODE)
 * (1-1) is the equation that has one independent variable and its derivatives with respect to the variable.
 * Highest order of derivative of (1-1) is 1 == > (1-1) is 1st Order Ordinary Differential Equation(ODE)
 * To be a Linear Equation, (1-1) must be satisfied with following two conditions
 * {| style="width:100%" border="0" align="center"
 * To be a Linear Equation, (1-1) must be satisfied with following two conditions
 * {| style="width:100%" border="0" align="center"
 * {| style="width:100%" border="0" align="center"

$$\displaystyle F(u+v) = F(u) + F(v) $$ $$\displaystyle (1-2) $$
 * }
 * }

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$$\displaystyle F(\alpha u) = \alpha F(u) $$ $$\displaystyle (1-3) $$
 * }
 * }

Step 1
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$$\displaystyle F(u+v) = M(x,u+v) + N(x,u+v)(u+v)' $$ $$\displaystyle (1-4) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(u) + F(v) = M(x,u) + N(x,u)u' + M(x,v) + N(x,v)v' $$ $$\displaystyle (1-5) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(u+v) {\neq} F(u) + F(v) $$ $$\displaystyle (1-6) $$
 * }
 * }

Step 2
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$$\displaystyle F(\alpha u) = M(x,\alpha u) + N(x,\alpha u)(\alpha u)' $$ $$\displaystyle (1-7) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \alpha F(u) = \alpha (M(x,u) + N(x,u)u') $$ $$\displaystyle (1-8) $$
 * }
 * }

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$$\displaystyle F(\alpha u) {\neq} \alpha F(u) $$ $$\displaystyle (1-9) $$
 * }
 * }

(1-6) and (1-9) ==> (1-1) is a Non-linear equation

 * {| style="width:100%" border="0" align="center"

 * style="width:92%; padding:10px; border:2px solid #8888aa" align="center" |
 * Therefore, (1-1) is a General Non-linear 1st Order Ordinary Differential Equation(N1-ODE)
 * Therefore, (1-1) is a General Non-linear 1st Order Ordinary Differential Equation(N1-ODE)

 * }
 * }
 * }

Solution of 2)

example :
 * {| style="width:100%" border="0" align="left"

$$\displaystyle xy + xy' = 0 =: F(y) $$ $$\displaystyle (1-10) $$
 * }
 * }

Step 1
 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(u+v) = x(u+v) + x(u+v)' $$ $$\displaystyle (1-11) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(u) + F(v) = xu + xu' + xv + xv' $$ $$\displaystyle (1-12) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(u+v) = F(u) + F(v) $$ $$\displaystyle (1-13) $$
 * }
 * }

Step 2
 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(\alpha u) = x(\alpha u) + x(\alpha u)' $$ $$\displaystyle (1-14) $$
 * }
 * }

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$$\displaystyle \alpha F(u) = \alpha (xu + xu') $$ $$\displaystyle (1-15) $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle F(\alpha u) = \alpha F(u) $$ $$\displaystyle (1-16) $$
 * }
 * }

(1-13) and (1-16) ==> (1-10) is a linear equation

 * {| style="width:100%" border="0" align="center"

 * style="width:92%; padding:10px; border:2px solid #8888aa" align="center" |
 * Therefore, (1-10) is an example of L1-ODE
 * Therefore, (1-10) is an example of L1-ODE

<Author> Egm6321.f10.team5.oh
 * }
 * }
 * }

= Problem #2 = Verify (7) from p. 7-1 is N1-ODE

Given

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\underbrace{(4x^7 + sin y)}_{M(x,y)} + \underbrace{(x^2y^3)}_{N(x,y)}y' = 0 =: F(y) $$
$$\displaystyle (2-1) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Find
 Verify (2-1) is N1-ODE

Solve
== > (2-1) is Ordinary Differential Equation(ODE)
 * (2-1) is the equation that has one independent variable and its derivatives with respect to the variable.
 * Highest order of derivative of (2-1) is 1 == > (2-1) is 1st Order Ordinary Differential Equation(ODE)
 * To be a Linear Equation, (2-1) must be satisfied with following two conditions
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 * To be a Linear Equation, (2-1) must be satisfied with following two conditions
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$$\displaystyle F(u+v) = F(u) + F(v) $$
$$\displaystyle (2-2) $$
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 * }

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$$\displaystyle F(\alpha u) = \alpha F(u) $$
$$\displaystyle (2-3) $$
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 * }

Step 1
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$$\displaystyle F(u+v) = 4x^7 + sin(u+v) + x^2(u+v)^3(u+v)' $$
$$\displaystyle (2-4) $$
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 * }

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$$\displaystyle F(u) + F(v) = 4x^7 + sin u + x^2u^3u' + 4x^7 + sin v + x^2v^3v' $$
$$\displaystyle (2-5) $$
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 * }

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$$\displaystyle F(u+v) {\neq} F(u) + F(v) $$
$$\displaystyle (2-6) $$
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Step 2
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$$\displaystyle F(\alpha u) = 4x^7 + sin(\alpha u) + x^2(\alpha u)^3(\alpha u)' $$
$$\displaystyle (2-7) $$
 * }
 * }

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$$\displaystyle \alpha F(u) = \alpha (4x^7 + sin u + x^2u^3u') $$
$$\displaystyle (2-8) $$
 * }
 * }

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$$\displaystyle F(\alpha u) {\neq} \alpha F(u) $$
$$\displaystyle (2-9) $$
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 * }

(2-6) and (2-9) ==> (2-1) is a Non-linear equation

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 * Therefore, (2-1) is a Non-linear 1st Order Ordinary Differential Equation(N1-ODE)
 * Therefore, (2-1) is a Non-linear 1st Order Ordinary Differential Equation(N1-ODE)

 * }
 * }
 * }

<Author> Egm6321.f10.team5.oh

= Problem #3 =

Solve
= Problem #4 =

Solve

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$$ F(x,y,y') = {\frac {d{\phi}(x,y)}{d x}} = {\frac} + {\frac} {\frac{dy}{dx}}$$

(4.1)
 *

 * }

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$$\begin{align} & M(x,y):={\phi}_{x}(x,y): =\frac{\partial \phi(x,y)}{\partial x} \\ & N(x,y):={\phi}_{y}(x,y): =\frac{\partial \phi(x,y)}{\partial y} \\ \end{align}$$

(4.2)
 *

 * }

So, that is
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$$F(x,y,y') = M\left(x,y\right)+N(x,y)y' $$

(4.3)
 *

 * }

From the given $${{\phi}\left(x,y\right)}$$, we can get M(x,y) and N(x,y)

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$$\begin{align} &M\left(x,y\right) :={\phi}_{x}(x,y)= 2 x y^{\frac{3}{2}} + {\frac{3}{x}}\\ &N(x,y) := {\phi}_{y}(x,y)= {\frac{3}{2}} y^{\frac{1}{2}}x^{2} + {\frac{2}{y}}\\ \end{align} $$      (4.4)
 *

 * }

Follow the $$ 1^{st} $$ condition of exactness,

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$$F(x,y,y') = M\left(x,y\right)+N(x,y)y' = \left( 2 x y^{\frac{3}{2}} + {\frac{3}{x}} \right) + \left( {\frac{3}{2}} y^{\frac{1}{2}}x^{2} + {\frac{2}{y}} \right) y' = 0 $$      (4.5)
 *

 * }

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$$ y' = - \frac{M\left(x,y\right)}{N(x,y)} $$      (4.6)
 *

 * }

And then, apply $$ 2^{nd} $$ condition of exactness,

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$$ \underbrace{M_{y}\left(x,y\right)}_{\frac{\partial M(x,y)}{\partial y}}= \underbrace{N_{x}(x,y)}_{\frac{\partial N(x,y)}{\partial x}} $$      (4.7)
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 *

 * }

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$$ {\frac {{\partial}M(x,y)} {\partial y} } = {\frac {{\partial} \left( 2 x y^{\frac{3}{2}}+{\frac{3}{x}}\right)} {\partial y}} = 3 x y^{\frac{1}{2}} = {\frac {{\partial}\left( {\frac{3}{2}} y^{\frac{1}{2}} x^{2} + {\frac{2}{y}}\right)} {\partial x}} = {\frac {{\partial} N(x,y)}{\partial x}} $$      (4.8)
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 *

 * }

So, F is exact N1_ODE.

And 3 more equations,

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$$\displaystyle {\phi}(x, y) = x^{5} + y^{3} = k $$ (4.9)
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$$ {\phi}(x, y) = x^{4} \ ln \ y = k $$ (4.10)
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 *

 * }

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$$ {\phi}(x, y) = {\frac{1}{4}}( x^{4} + {6} x^{2} y^{2} + y^{4})= k $$ (4.11)
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 *

 * }

= Problem #5 =

Solve
= Problem #6 =

Solve
Total differential of function $$\phi\left(x,y\right)$$,

As$$\phi\left(x,y\right)=k=const$$, then $$ d\mathbf{\phi}=0 $$, We can rewrite the equation 6.5,

For the given problem,

So, we can verify that the equation 6.1 could be converted to the form of the first condition of exactness, equation 6.2.

The mixed derivatives of $$\begin{align}\phi\end{align}$$, $$\begin{align}\phi_{xy},\phi_{yx} \end{align}$$ are both zero, meeting the second condition of exactness. This is clearly seen from equation 6.7 as $$\begin{align}\phi_{x} \end{align}$$ is only a function of $$\begin{align}x\end{align}$$ while $$\begin{align} \phi_{y} \end{align} $$ is just a function of $$\begin{align}y \end{align}$$.

= Problem #7 = '''Solving Non-Homog. L-1-ODE-VC'''

Given

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$$\displaystyle a_1(x)y' + a_0(x)y = b(x) $$
$$\displaystyle (7-1) $$
 * }
 * }

Find

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1) Assume $$\displaystyle a_1(x) = 1, a_0(x) = x, b(x) = 2x + 3, $$ Find $$\displaystyle h(x) $$ and $$\displaystyle y(x) $$

2) Assume $$\displaystyle a_1(x) {\neq} 0, \ \forall x$$ (7-1) becomes $$\displaystyle y' + \underbrace{\frac{a_0(x)}{a_1(x)}}_{P(x)}y = \underbrace{\frac{b(x)}{a_1(x)}}_{Q(x)},  $$ Find $$\displaystyle y(x)  $$ in terms of $$\displaystyle a_0, a_1, b  $$
$$\displaystyle (7-2) $$
 * }
 * }

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3) Assume $$\displaystyle a_1(x) = x^2 + 1, a_0(x) = x, b(x) = 2x, $$ Find $$\displaystyle h(x) $$ and $$\displaystyle y(x) $$
 * }
 * }

Solve
According to Lecture(Mtg 10. Tue, 14 Sep 10), (5) p.10-2, (1) p.10-3 and (6) p.10-3 We can rewrite,

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y' + a_0(x)y = b(x) $$
$$\displaystyle (7-3) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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h(x) = exp[\int^{x} a_0(s) ds] $$
$$\displaystyle (7-4) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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y(x) = \frac{1}{h(x)} \int^{x} h(s)b(s) ds] $$
$$\displaystyle (7-5) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Solution of 1)
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h(x) = exp[\int^{x} s ds] $$
$$\displaystyle (7-6) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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$$h(x) = exp(\frac{1}{2} x^2)$$
$$\displaystyle (7-7) $$
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 * }
 * }

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y(x) = \frac{1}{exp(\frac{1}{2} x^2)} \int^{x} exp(\frac{1}{2} s^2)(2s + 3) ds] $$
$$\displaystyle (7-8) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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y(x) = \frac{1}{exp(\frac{1}{2} x^2)} [\underbrace{2 \int^{x} exp(\frac{1}{2} s^2)s \ ds}_{<1>} + 3 \int^{x} exp(\frac{1}{2} s^2) ds] $$
$$\displaystyle (7-9) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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<1> \ t := \frac{1}{2} s^2 $$
 * $$\displaystyle
 * $$\displaystyle

$$\displaystyle dt = s \ ds, \ ds = \frac{dt}{s} $$

$$\displaystyle <1> \ --> \ 2 \int^{x} exp(t) dt $$

$$\displaystyle = 2 exp(\frac{1}{2} x^2) $$
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 * }
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$$\therefore \ y(x) = 2 + \frac{3}{exp(\frac{1}{2} x^2)} \int^{x} exp(\frac{1}{2} s^2) ds $$
$$\displaystyle (7-10) $$
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 * }
 * }

Solution of 2)

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$$h(x) = exp[\int^{x} \frac{a_0(s)}{a_1(x)} ds] $$
$$\displaystyle (7-11) $$
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 * }
 * }

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$$y(x) = \frac{1}{exp[\int^{x} \frac{a_0(s)}{a_1(x)} ds]} \int^{x} exp[\int^{x} \frac{a_0(s)}{a_1(x)} ds] \ \frac{b(s)}{a_1(s)} ds $$
$$\displaystyle (7-12) $$
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 * }

Solution of 3)

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h(x) = exp[\underbrace{\int^{x} \frac{s}{s^2 + 1} \ ds}_{<2>}] $$
$$\displaystyle (7-13) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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<2> \ t := s^2 + 1 $$
 * $$\displaystyle
 * $$\displaystyle

$$\displaystyle dt = 2s \ ds $$

$$\displaystyle \therefore ds = \frac{1}{2} \frac{dt}{s} $$

$$\displaystyle h(x) = exp[\int^{x} \frac{1}{2} \frac{1}{t} dt] $$

$$\displaystyle h(x) = exp(\frac{1}{2} log (x^2 + 1) ) $$
 * }
 * }
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$$h(x) = \sqrt{ \left(x^2+1\right) } $$
$$\displaystyle (7-14) $$
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 * }
 * }

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y(x) = \frac{1}{\sqrt{ \left(x^2+1\right)}} \int^{x} \sqrt{ \left(s^2+1\right)} \frac{2s}{s^2 + 1} ds $$
$$\displaystyle (7-15) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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y(x) = \frac{1}{\sqrt{ \left(x^2+1\right)}} \underbrace{\int^{x} \frac{2s}{\sqrt{ \left(s^2+1\right)}} \ ds }_{<3>} $$
$$\displaystyle (7-16) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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<3> \ u := s^2 + 1 $$
 * $$\displaystyle
 * $$\displaystyle

$$\displaystyle du = 2s \ ds, \ ds = \frac{dt}{2s} $$

$$\displaystyle <3> \ --> \int^{x} \frac{1}{\sqrt{ \left(u \right)}} du $$

$$\displaystyle = 2 \sqrt{ \left(x^2 + 1 \right)} $$
 * }
 * }
 * }

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$$\therefore \ y(x) = 2 + \underbrace{\frac{k}{\sqrt{ \left(x^2 + 1 \right)}}}_{k \ : \ int. \ const} $$
$$\displaystyle (7-17) $$
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 * }
 * }

<Author> Egm6321.f10.team5.oh

= Problem #8 =

Solve
= Problem #9 =

title: User talk:Egm6321.f10.team5.oh/hw2