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Motivation and emotion/Book/2010/Violent crime motivation

Overview
'“The problem of crime is the problem of the criminal mind. An understanding of criminal behavior can only be reached by a study of the criminal consciousness …”' Editorial, April-June, 1928 Violence has been present throughout the ages, since the dawn of man when caveman fought one another for food and access to females, through to ancient times when empires were created through bloodshed or conquered through ultimate force, to medieval times when wars were waged in the name of religion, through to modern times with the rise of the serial killer. The motivation behind why people act in violent and destructive ways towards one another is a question that has been contemplated over the centuries and it is one we shall attempt to shed some more light on in this chapter. It is usually the socio-cultural aspects that are looked at in relation to violent and criminal behaviour – an individual’s upbringing, the sort of father figure they had, early family life, education process, etc. and as such, the social determinants of violent and criminal behaviour have been looked at extensively already.  "One side of me says, I'd like to talk to her, date her. The other side of me says, I wonder what her head would look like on a stick?"
 * Ed Kemper

We shall take a closer look at the biological and personality features of violent behaviour, which have only just begun to come more to the forefront within the past decade or so. After exploring those two topics we shall move on to the motivation behind single and serial homicides, an interesting and disturbing subject, especially in relation to serial homicide – what would make a person want to kill multiple other people, often in cruel ways? This question has plagued psychologists, law enforcement personnel and the general public since Jack the Ripper stepped out on to the streets of London in 1888. Hopefully after reading this chapter you may better be able to formulate your opinion on why these people did the things that they did, and why others continue to perform this horrific behaviour.

Aggression

“A feeling of hostility that arouses thoughts of attack.” Aggression Definition

“Violent action that is hostile and usually unprovoked.” Aggression Definition

“In psychology, as well as other social and behavioural sciences, aggression (also called combativeness) refers to behaviour between members of the same species that is intended to cause pain or harm.” Aggression Definition

“Acts or threats designed to cause injury.” Aggression Definition

“Forceful action against another person which may be physical, verbal, or symbolic, and is meant to cause pain. Such behaviour may be hostile or destructive or it may be for self-protection.” Aggression Definition

“Overt or suppressed hostility, either innate or resulting from continued frustration and directed outward or against oneself.” Aggression Definition

What these definitions do not show, however, is that aggression is a part of normal every day life. The problem in regards to the common understanding of aggression is that over the years the term ‘aggression’ has taken on the implication of harm - that if someone is showing aggressive behaviour, by extension, they must be harming someone. This is not the case. Everyone experiences aggression in his or her daily life; people act aggressively in pursuing goals or in trying to accomplish something, people may feel aggression but not act on it or they may channel it in to something purposeful (Abrams, n.d.). When discussing violence and the motivations behind why people commit violent acts the distinction must be made, and it must be kept in mind, that to perform a violent act ‘requires’ aggression, but aggression does not always lead to violence (Abrams, n.d.; Perry, n.d.). For the purposes of this chapter though, only the form of aggression that leads to violence shall be focussed on. Aggressive behaviour displayed in adults can be sub-typed into four different categories on the basis of the motivation that drives it:  "I have an obsession with the unattainable. I have to eliminate what I cannot attain."
 * Robert Bardo.

 * With intent to destroy the victim
 * Executed during a crime
 * Acted upon during an episode of intense rage or anger
 * Considered as psychopathic
 * (Hinde, 1987)

Each of these motivations may come into play during an act of violent crime.

Violence

“Extreme force; Action intended to cause destruction, pain, or suffering; Widespread fighting; Injustice, wrong.” Violence Definition

“A turbulent state resulting in injuries and destruction.” Violence Definition

“Physical force used so as to injure, damage, or destroy; extreme roughness of action.” Violence Definition

“Unjust or callous use of force or power, as in violating another's rights, sensibilities, etc.” Violence Definition

“Brutality, bloodshed, savagery, fighting, terrorism, frenzy, thuggery, destructiveness, bestiality, strong-arm tactics (informal), rough handling, bloodthirstiness, murderousness.” Violence Definition

"The intentional use of physical force or power, threatened or actual, against oneself, another person, or against a group or community, that either results in or has a high likelihood of resulting in injury, death, psychological harm, maldevelopment, or deprivation." Violence Definition

Violence is a complex construct, made up of and motivated by many different factors. One way violence can be categorised is by how the violence is inflicted upon the victim:  "You feel the last bit of breath leaving their body. You're looking into their eyes. A person in that situation is God!"
 * Ted Bundy.

 * 1) Physical violence
 * 2) Sexual violence
 * 3) Psychological violence
 * 4) Deprivation
 * (World Health Organisation, n.d.)

The second way of categorising violence is with regard to the connection between the perpetrator and the victim:
 * Self-directed, which is violence committed by an individual against themselves. Consists of self-harm and suicide.
 * Interpersonal, which is violence between individuals. There are two further categories contained within interpersonal violence –
 * Family and intimate partner violence. Consists of elder abuse, child abuse/maltreatment and domestic violence.
 * Community violence. Made up of stranger violence and acquaintance violence. Consists of assault, workplace violence, youth violence, and violence whilst committing a crime.

 * Collective violence, which is violence committed by groups of people. Consists of political violence, economic violence and social violence.
 * (World Health Organisation, n.d.)

Motivation

“The psychological feature that arouses an organism to action toward a desired goal; the reason for the action; that which gives purpose and direction to behaviour.” Motivation Definition

“Motivate - give an incentive for action.” Motivation Definition

“Motivation is the activation or energisation of goal-oriented behaviour.” Motivation Definition

Motivation is an important concept in psychology. Psychologists (and the general public!) are constantly seeking the answer to why people do the things they do – what it is that motivates them to perform certain behaviours or react in specific ways to stimuli. Motivation is relevant in many spheres of psychology: social psychology (e.g. how do social roles and group memberships motivate an individual’s behaviour?), abnormal psychology (e.g. how does having a mental disorder [depression, schizophrenia, mania] affect a person’s motivation), personality (e.g. does having an extroverted personality compared to an introverted personality influence motivation?), intelligence (does motivation differ between intelligence levels?), cognitive (can motivation be measured in the brain?), emotion (which emotions are more likely to motivate a person?), physiological (fight or flight reaction), learning (what motivates a person to learn?), and many more. Perhaps one of the most significant areas in which motivation is applicable, and extremely important, is in criminology and the study of criminal behaviour.

Violent crime
Violent and aggressive behaviour is a constant problem in society; one needs only to turn on the nightly news to be confronted with images and stories of violence happening around the world. In America 1,318,398 violent crimes were committed in the year 2009 (Crime in the United States, 2009). The Federal Bureau of Investigations clusters four crimes into the violent crime category:
 * 1) Murder and nonnegligent manslaughter
 * 2) Aggravated assault
 * 3) Forcible rape
 * 4) Robbery
 * (Crime in the United States, 2009)

According to the Uniform Crime Reporting (UCR) Program run by the FBI, a violent crime is “those offenses which involve force or threat of force” (Crime in the United States, 2009). The definitions from the UCR Program of the first three of the above violent crimes is included below (for the purposes of this chapter robbery shall not be focussed on), so that the reader may more easily comprehend the implications of these crimes and will be able to better understand what it is that we shall be referring to throughout the rest of this chapter. Knowledge of the type of crime committed will also assist in understanding the motive behind why it was committed.

Murder and nonnegligent manslaughter: “the willful (nonnegligent) killing of one human being by another. The classification of this offense is based solely on police investigation as opposed to the determination of a court, medical examiner, coroner, jury, or other judicial body. The UCR Program does not include the following situations in this offense classification: deaths caused by negligence, suicide, or accident; justifiable homicides; and attempts to murder or assaults to murder.” (Crime in the United States, Murder, 2009)

Aggravated assault: “an unlawful attack by one person upon another for the purpose of inflicting severe or aggravated bodily injury. The UCR Program further specifies that this type of assault is usually accompanied by the use of a weapon or by other means likely to produce death or great bodily harm. Attempted aggravated assault that involves the display of—or threat to use—a gun, knife, or other weapon is included in this crime category because serious personal injury would likely result if the assault were completed.” (Crime in the United States, Aggravated Assault, 2009)

 "If I killed them, you know, they couldn't reject me as a man. It was more or less making a doll out of a human being . . . and carrying out my fantasies with a doll, a living human doll."
 * Edmund Kemper.

Forcible rape: “the carnal knowledge of a female forcibly and against her will. Attempts or assaults to commit rape by force or threat of force are also included; however, statutory rape (without force) and other sex offenses are excluded.” (Crime in the United States, Forcible Rape, 2009)

In 2009 there were approximately 806,843 aggravated assaults, 88,097 rapes and 13,636 murders committed in America (Crime in the United States, Aggravated Assault, Homicide, Rape, 2009). In regards to the homicide data, 77% of the victims were male, 89.7% of the offenders were male, 71.8% of homicides involved firearms, 34.6% of female victims were murdered by their husband/boyfriend, 24.2% of all victims were killed by a family member, 53.8% were killed by someone they knew, 41.2% were killed during an argument, and 22.9% were killed during the committal of a felony offence (Crime in the United States, Expanded Homicide data, 2009).

The tangled web of violence

What causes violence? Are people born hard-wired to become criminals? Or does their environment and social world shape them into deviants? Is there an ‘aggression’ gene? Do violent videogames/movies/TV shows/lyrics model and encourage violence? Does maternal neglect or lack of a ‘father figure’ produce criminal behaviour? Does abuse beget violence? Does lower intelligence promote criminality? Peer pressure? Hormone levels? Seasonal changes? Personality types/characteristics? Each of these theories corresponds to a different area within the boundaries of the psychology discipline. We shall first take a look at the biological side of violence and the theories used to explain the motivations behind criminal behaviour before moving on to the theories related to personality and violence.

The biology of violence, is it in the genes?
Criminology has tended not to focus on the neurological and genetic factors of violent behaviour, instead choosing to concentrate on the social influences that contribute to criminality (Tehrani & Mednick, n.d.). However, within about the past two decades there has been an emergence in the acceptance and implementation of a biological theory of aggression that has turned the debate on the motivation behind violence from a social influences perspective to a more biological perspective. Though the nature/nurture debate continues to rage, there has been a steady acknowledgement that both nature and nurture play a part in what traits an individual possesses and how these traits are expressed (Beaver & Holtfreter, 2009). In this section we shall focus on the “nature” part of the debate in relation to violent behaviour.

Brain areas related to violence
Violence, at its basic level, can be categorised into two distinct types – premeditated or impulsive (Davidson, Putnam & Larson, 2007; Volavka, 1999). An interesting study that supports this division and also points to a neurological basis of violence looked at the effect of administering the drug phenytoin to prisoners, phenytoin diminished the instances of impulsive violence but not premeditated violence, suggesting that the different types of aggression may be located in different parts of the brain (Barratt, Stanford, Felthous & Kent, 1997). Studies that have focussed on the heritability of aggression have proposed that up to 50% of the variability in aggression is determined by an individual’s genes (though these estimates range from 40% all the way up to 80% depending upon the age cohort that was sampled) (Brendgen et al., 2005). With around 50,000 diverse genes estimated to be expressed in the brain of any individual, it can seem a bit like trying to find a needle in a haystack when it comes to discovering which brain areas and specific genes are involved in aggressive behaviour (Feldker, Kloet, Kruk & Datson, 2003). But through a variety of different techniques, such as brain stimulation, lesioning of specific brain areas, neuronal tracing and injections of hormones, transmitters and drugs, an outline of the major brain areas responsible for violent behaviour has emerged (Feldker et al.). As aggression is not homogeneous between all human beings it is unfeasible to attempt to narrow it down into only a few brain structures and neural networks, numerous studies have uncovered a number of aggression ‘hotspots’ in the brain:
 * The amygdala (fear and aggression responses)
 * The cortico-medial part of the amygdala has been showing promising signs of being related to excessive acts of aggression

 * Lateral hypothalamus (prey aggression)
 * Medial hypothalamus (affective aggression)
 * Dorsal hypothalamus (flight or fight response)
 * (Birbaumer & Flor, 1998)

Evidence points to the pre-frontal cortex, the amygdala and the temporal lobe being the prime mediators of violent behaviour, as they are entrusted with producing socially acceptable interactions, and it is when these regions are damaged or impaired in some way that aggressive behaviour manifests itself (Birbaumer & Flor, 1998). As an example, the pre-frontal cortex (PFC) of 41 murderers was examined, the researchers found that the lateral and medial regions of the PFC were under-activated and the right amygdala was over-activated, the impulsive murderers also exhibited a decline in lateral PFC metabolism and an amplified metabolism in the amygdala, hippocampus, thalamus and midbrain compared to the premeditated murderers (Davidson, Putnam & Larson, 2007; Volavka, 1999). These are significant results because the PFC is an important component of emotional regulation in the brain and has been linked to aggressive behaviour in other studies, as well as having a large concentration of serotonin receptors (Davidson, Putnam & Larson, 2007).

Serotonin and violence
Certain neurotransmitters have also been found to play a part in aggressive behaviour; specifically serotonin (5-hydroxytryptamine; 5-HT) has been shown to have an inhibitory effect in regards to impulsive violence (Davidson, Putnam & Larson, 2007; Lowenstein, 2003; Retz, Retz-Junginger, Supprian, Thome & Rösler, 2004; Volavka, 1999). Research done with psychiatric patients who suffered from a range of mental disorders has shown that complications in the performance of 5-HT is correlated with poor impulse control, suicidality and aggression directed at other individuals (Beaver, 2009; Retz et al., 2004). Pharmacological challenge tests and platelet binding studies have also supported the association between violence and a reduction in serotonin functioning, and experiments looking at the level of 5-hydroxyindolacetic acid (what serotonin is converted into) in cerebrospinal fluid in criminal delinquents, psychiatric patients and individuals who committed suicide by violent means also provides evidence for the association between low serotonin levels and violent behaviour (Davidson, Putnam & Larson, 2007; Retz et al., 2004; Wood, 1998). A study conducted by Retz et al. (2004) looked at the serotonin transporter gene in 153 adult male volunteers (of which 132 were criminal defendants) and found that there was a considerable relationship between violent behaviour and a variation of the 5-HT transporter gene. Those volunteers who had a short (s) variant of the 5-HT transporter gene allele showed substantially higher instances of violent behaviour, which is also corroborated by other studies that found that violent behaviour displayed by alcoholics and violent suicide was also linked with the s allele (Retz et al., 2004).

The biology of violence is an important factor in understanding criminal and aggressive behaviour, only a mere sliver of the research that has been conducted in recent years has been touched on here, as the topic of the genetics and neurochemistry involved in aggression and criminal behaviour is such a vast area that one would need a whole chapter to even give the subject a proper overview. However, hopefully this section has made clear that a person can be genetically predisposed to commit violent acts, though this is no excuse for aggressive behaviour it demonstrates that the motivation behind violence can, in part, be biological and not just psychological and social.

A violent and criminal personality
 "I always had the desire to inflict pain on others and to have others inflict pain on me. I always seemed to enjoy everything that hurt. The desire to inflict pain, that is all that is uppermost."
 * Albert Fish

Criminality theories usually diverge in the way they view criminal behaviour. One section of theories sees crime as a collection and culmination of criminal activities, another set of theories looks at crime as an explicit event or action, and another division of theories views criminality as a psychological attribute or an aspect of an individual’s personality that produces an inclination for them to exhibit criminal behaviour (Gudjonsson & Sigurdsson, 2004). It is the latter take on criminality that we shall focus on in the next part of this chapter.

The Diagnostic and Statistical Manual of Mental Disorders 4th edition (DSM-IV-TR) defines personality traits as “enduring patterns of perceiving, relating to, and thinking about the environment and oneself that are exhibited in a wide range of social and personal contexts.” It seems reasonable to assume that an individual’s personality would play a role in determining the motives behind commiting certain forms of violence (Gudjonsson & Sigurdsson, 2004) and it has been suggested by many researchers that aggression, and the motivation to commit aggressive acts, should be looked at and dealt with as a facet of an individual’s personality makeup (Schlesinger, 1998). Douglas and Olshaker said in their book ‘The Anatomy of Motive’ (2000) that “behaviour reflects personality.” (p. 19)

The motivation to engage in violence and its relation to personality can be conceptualised in a few different ways. Farrington (Gudjonsson & Sigurdsson, 2004) described crime as the result of a process developing over four phases. Motives come into play in the first phase, where an individual’s desire for obtaining something, be it physical as in material goods or money, or psychological as in status, a sense of power or excitatory stimulation. In the second phase the would-be offender chooses an unlawful or morally unacceptable method in order to achieve their preferred outcome. In the third stage the motivation to commit the specific crime strengthens and intensifies, this can be due to a number of reasons – insufficient coping mechanisms, substandard moral development, a predisposition to react in a certain way to certain events and so forth. The last stage is the culmination of the criminal process, where an opportunity has presented itself and a decision has been made to go ahead and commit the crime, in the hopes of attaining the wanted object (or state of mind). Though it may not seem apparent at first glance, at each stage of this conceptualisation of criminality an individual’s personality comes into play. Specific personality traits can influence what feelings or items motivate a person to engage in criminal activity, and it can also influence the method they choose to employ in obtaining the desired physical object or psychological state of mind.

A second way of looking at criminality and personality is by matching thinking styles of criminals with specific personality traits. Using the Psychological Inventory of Criminal Thinking Styles, the personality traits of neuroticism, sensation-seeking and attention deficits have been found to be associated with criminal cognitions. A principal components analysis of the aforementioned scale revealed two main components – willful hostility and lack of thoughtfulness (Gudjonsson & Sigurdsson, 2004).

One can also seek to understand criminality in relation to personality disorders such as narcissism, paranoia and antisocial personality disorder or psychopathy (Coid, 2002; Loper, Hoffschmidt & Ash, 2001; Schlesinger, 1998). The DSM-IV-TR defines a personality disorder as “an enduring pattern of inner experience and behaviour that deviates markedly from the expectations of the individual’s culture and is manifested in at least two of the following areas: cognition, affectivity, interpersonal functioning, or impulse control.” We shall quickly take a look at narcissism and antisocial personality disorder/psychopathy and how they may motivate a person to behave violently and criminally.

Antisocial Personality Disorder
Antisocial Personality Disorder (ASPD), also termed psychopathy and sociopathy, is a personality disorder characterised by a blatant disrespect for and violation of the rights of others, a distinct lack of empathy, no discernible regret or shame shown for the negative consequences of their actions, a failure to conform to government laws and social norms, a dishonest and manipulative nature, poor impulse control, aggressive behaviour and the committal of violent acts against others, show little regard for their own safety and the safety of others and often engage in high-risk behaviours (American Psychiatric Association, 2000, Diagnostic and statistical manual of mental disorders). Individuals with ASPD have been shown to be more likely than other members of a population to die prematurely and in a violent way, and one of the associated features of the disorder is that they may spend many years in a penal institution; ASPD is often shown to feed into the construct of the ‘career criminal’ (Coid; DSM-IV-TR). Psychopathy is connected to instrumental violence; meaning that the violence is performed with a preplanned goal in mind that is generally indulging some ‘need’ felt by the perpetrator (Loper, Hoffschmidt & Ash, 2001). As such, a lot of serial killers are labeled as psychopaths or sociopaths.

Narcissistic Personality Disorder
Narcissistic Personality Disorder (NPD) is similar in some ways to ASPD; individuals suffering from NPD display a sense of grandiose self-importance and superiority to others, they believe they are unique and special and expect others to identify them in this way, they often fantasise about power, intelligence, beauty and infinite success, they feel they are entitled to privileged and favourable treatment, they have an inflated self-esteem/ego and need constant admiration and positive attention, and they lack empathy towards others, failing to realise that other people have feelings and are often contemptuous towards others who talk about their own problems (DSM-IV-TR). ASPD often shows features of NPD and NPD can be associated with ASPD, so it would not be surprising to come across an individual who had ASPD and also meet the criteria for NPD (DSM).

Coid's prison research (2002)
 "I didn't want to hurt them, I only wanted to kill them."
 * David Berkowitz

To tie these personality disorders to the motivations behind violent behaviour, Coid (2002) performed a study looking at the association between motivations to perform violent and disorderly behaviour in prison and having an axis II personality disorder (which ASPD and NPD qualify as). Eighty-one male prison inmates took part in the research, with 84% of them classed as having ASPD, and 73% of that number classed as psychopaths. 63% had NPD, with there being a high co-morbidity rate between the two disorders. Coid found that the prisoners who were classified as psychopathic and narcissistic were more often motivated by feelings of pride (in their fighting proficiency and physical capability), a perceived attack to their self-worth and a subsequent lowering of their self-esteem, and the belief that violence is the sole answer to addressing and solving interpersonal difficulties. They were also less likely to conform to prison regulations, follow rules set by the guards and were relentless in their attempts at insubordination. Those participants who displayed NPD were also more likely to exhibit violence towards prison staff, which can be attributed to their feelings of superiority over all others (including the prison guards) and that they are entitled to privileged treatment, which, of course, they would not receive. As Coid stated, “Major discrepancies between their expectations of how they should be managed and the reality of prison discipline led to many serious behavioural problems”. Narcissism has also been correlated to an offender’s recidivism rate, as one of the indicators of narcissism is dehumanisation of other people, the amount that a murderer dehumanises their victim helps predict the likelihood of them reoffending – the higher the dehumanisation (and the higher the narcissism) the more likely the individual is to become a serial murderer (Schlesinger, 1998).

Gudjonsson and Sigurdsson's personality traits and offender motivation study (2004)
Gudjonsson and Sigurdsson (2004) also looked at personality traits and the motivations for offending, this time among young offenders. They had two samples of participants – 382 participants from an Iceland college with an average age of 19, and 380 undergraduate students from an Iceland university with an average age of 24, for a total sample of 762. Participants’ self-reported delinquent acts that they had committed during the past year and were instructed to choose one serious offence to keep in mind when they completed a number of psychological scales, but we shall focus on the Offence Motivation Questionnaire (OMQ) as it relates more closely to personality. Crimes were divided into traffic offences, theft, violent offences, drug offences, deception, and damage to property. The results from the OMQ were put into a factor analysis and five motivational factors were revealed: 73% of the overall sample was found to have committed their most serious offence while in the company of others, pointing to the influence of peers on the offending behaviour of adolescents, however, this is only a strong influence when the individual has the personality trait of being compliant, and as such offended to try and please their peer group. Gudjonsson and Sigurdsson (2004) found that antisocial personality traits were correlated with the provocation, excitement and financial motivations, and particularly, in regards to the provocation motivation, with taking revenge upon someone for a perceived slight. A person with antisocial personality traits is also more likely to disregard or ignore the consequences of their actions, which can be due to them either being unconcerned with what the consequences are or because they act recklessly and with little forethought (Gudjonsson & Sigurdsson).
 * 1) Excitement
 * 2) Provocation
 * 3) Compliance
 * 4) (failure to consider) Consequences
 * 5) Financial

Once again, there are many different facets to an individual’s personality and this chapter does not cover them all. This section simply aims to give an introduction to how personality can interact with the motivation to perform violent or criminal behaviour.

Identifying the motive behind a crime
 Name: Jeffery Dahmer Nickname: “The Milwaukee Cannibal” “The Milwaukee Monster” Birth: May 21 1960 Death: November 28 1994 [severe head trauma – killed by an inmate in prison] Number of victims: 17 Method of killing: strangulation Active: 1978-1991

Identifying the motive behind a crime (especially a violent crime) is important for two main reasons: Motive, in terms of criminology and law, “combines personal experiences and reasons with the actual situation and with future states of the person” (Kroeber, 2007); motive is the result of the merging together of psychological justification and purpose (Kroeber, 2007). Police officers would be some of the first people to tell you that crimes are motivated by any number of reasons, the ‘traditional’ motives being sex, financial gain, jealousy, love, revenge, elimination of a rival and obsession (Brantley & Kosky, 2005). With the rise of the serial killer in the twentieth century, criminologists realized that they would have to develop a set of more multifaceted and elaborate motives to explain the reasons behind why an individual would commit multiple acts of murder, usually extremely violent and heinous ones (Brantley & Kosky, 2005). Serial killers are typically driven through some form of internal desire that they feel compelled to satisfy each time it arises, a serial murderer is more often motivated by feelings of control, power, domination and mastery that they gain from the killing act and the satisfaction they feel upon completion of the act (Brantley & Kosky).
 * 1) The prosecution must prove that the defendant in a case had criminal intent, also called mens rea, meaning ‘guilty mind’ (Barratt & Felthous, 2003). For an act to be judged as criminal, the person who committed it must be shown to have had criminal purpose to be convicted, in other words, they must be shown to have had motive (Barratt & Felthous, 2003).
 * 2) To understand the reason/s behind why a person would perform such a heinous and immoral act against another human being or group of human beings. By recognising the causes behind violent behaviour it can be hoped that other acts similar to it can be prevented (editorial, 1928).

Schlesinger (2004) proposed a framework in an attempt to explain the different aspects of criminal motivation. His theory located criminal motivation along a continuum with external motivators from an individual’s social and environmental surroundings to internal motivators that are due to an individual’s psychological and biological makeup.  Name: John Wayne Gacy Nickname: “The Killer Clown” Birth: March 17 1942 Death: May 10 1994 [lethal injection] Number of victims: 33 Method of killing: generally suffocation, occasionally stabbing Active: 1972-1978

From the external end to the internal end the spectrum separates criminal motivation into environmental, situational, impulsive, catathymic, and compulsive. Schlesinger (2004) stated that crimes that are motivated by a person’s environment often include offences committed by people in a group such as gang rape, arson or property damage during a riot, and more famously the Manson family murders and the atrocities committed under the Nazi command. The experiments performed by Asch and Milgram demonstrated that ordinary, everyday people could, and would, conform to a group consensus and obey a person in authority even if they knew that what they were conforming to or obeying was wrong, either practically or morally (remember Gudjonsson and Sigurdsson’s research with adolescent offenders as well).

Situational motivators account for around 70% of all homicides, most murders occur during an argument or the committal of another offence when the perpetrator is experiencing a large amount of pressure and anxiety, and as such can be classified as impulsive and unplanned (Schlesinger, 2004). The rate of repeat offenders in this category is very low since the homicide is ‘spur of the moment’ and involves no planning out. Crimes characterised by impulsive motivations are often poorly organised and executed, with scarcely any evidence of forethought, the impulsive offender usually has trouble regulating their emotions so they instead translate them into actions, which results in violent behaviour (Schlesinger). As one can imagine, the recidivism rate in this category is quite high and impulsive offenders generally commit multiple criminal acts. Maier came up with the term catathymia to describe the substance of delusions, before it was subsequently incorporated into criminological theory as an alternative basis for homicide – catathymia could be used to describe the “nondelusional changes in thinking that result in extreme acts of violence”, where an individual begins to think that violence is the only answer to a particular situation (Holcomb & Daniel, 1988; Schlesinger, 2004). Offenders who have experienced catathymia at the commencement of a felony have said that their state of mind after the act is restored back to its calm and logical self (Schlesinger, 2004). At the far end of the continuum lie the compulsive motivators, motivators that are completely psychological and internal and feel little influence from social or situational factors. This form of motivators is comparable to obsessive-compulsive disorder in that those who experience it feel tension and distress until they have carried out the act, after which they feel a sense of relief (Schlesinger). This form of offender usually carries out their crime in a ritualistic and highly planned manner, and due to the compulsion felt to commit a particular crime they often reoffend multiple times, however, if they come across a suitable victim and there is an opportunity then they will take it. Serial killers often fall into this category of motivators since their crimes are often highly ritualistic, carefully planned, are preceded by fantasies of carrying out the felony and, once they become a serial occurrence, have similarities across all offences (Schlesinger).

This spectrum motivation theory is useful in that it neatly encompasses the general range of expression that motivation may take when it comes to violent and criminal behaviour, though it may only be one theory it is comprehensive and broad enough to be used as a good introductory theory to the different styles of criminal motivators.

Homicide and serial homicide
 "My first murder was thrilling because I had embarked on the career I had chosen for myself, the career of murder."
 * John Reginald Halliday Christie

Homicide is the intentional killing of one person by another; serial homicide on the other hand seems to be a much harder crime to define. Serial homicide is often called serial murder, serial killing, multicide killing and multiple murder, which are all technically correct, however it is also called mass murder or spree murder, which are technically incorrect (Douglas & Olshaker, 2000). Former FBI agent and author of many books dealing with the criminal mind, John Douglas defines and differentiates between the terms of serial killer, mass killer and spree killer in this way:

"A serial killer is hunting human beings for the sexual thrill it gives him and will do it over and over again, believing he can outwit and outmanoeuvre the police, never expecting to be caught. The spree killer kills a number of victims at different locations in a short period of hours or days. But a mass killer is playing an endgame strategy. Once he commits himself to his course of action, he does not expect to come out of it alive. (The anatomy of motive, pp. 14-15)"

Homicide itself can be classified depending upon the offender, the victim, situational circumstances and, of course, the motive (Miethe & Drass, 1999). Some examples:  Name: Ted Bundy Nickname: N/A Birth: November 24 1946 Death: January 24 1989 [electrocuted] Number of victims: confessed to 30, estimates up to 100 Method of killing: strangulation Active: 1974-1978 [http://investigation.discovery.com/investigation/quiz-central/criminals/ted-bundy.html How much do you know about Ted Bundy? Click here to find out!]
 * First/second/third degree murder

 * Gang murder
 * Child victim
 * Adolescent victim
 * Female victim
 * Elderly victim
 * Domestic homicide
 * Workplace homicide
 * School homicide
 * Drive-by shootings
 * Road rage
 * Murder of a family member
 * Murder of an intimate partner
 * Murder of a stranger
 * Murder for revenge
 * Murder for financial gain
 * Murder out of jealousy
 * (Miethe & Drass, 1999)

The motivational aspect of homicides (both single and serial) we shall look at in more depth.

As we do, see if you can spot if there are any consistencies across the theories behind the motivations for serial homicide. If so, why do you think that particular motivation/s is included in the majority of the theories?

Instrumental/Expressive motives
 Name: Richard Ramirez Nickname: “The Night Stalker” Birth: February 28 1960 Death: awaiting execution Number of victims: over 30 Method of killing: shooting Active: 1984-1985

One way in which criminologists distinguish between types of motive is labelling them as either instrumental or expressive in nature (Miethe & Drass, 1999). Expressive killings are spur-of-the-moment and are heavily influenced by the current emotions of the person and are connected to impulsive violence, whereas instrumental killings are done to obtain a goal and are connected to premeditated violence (Miethe & Drass, 1999; Santtila et al., 2008). Common motivators for murders are altercations during drug deals, romantic triangles, jealousy, burglary, robbery, assault (sexual and nonsexual) and revenge (Miethe & Drass, 1999; Santtila et al.). It is widely accepted that serial homicides are almost always motivated by a psychological impulse or compulsion to kill, and usually have a sexual component to it, even if there is no sexual assault contained within the crime (Kraemer, Lord & Heilbrun, 2004). But researchers caution that the motivation behind serial murders is often abnormal, unusual and highly individualised; unlike single homicides, which can usually be grouped together depending on the motive behind them, the motives behind serial homicides are rarely similar and often differ significantly from one offender to the next (Knight, 2006; Santtila et al., 2008). There has also been a lack of empirical evidence in regards to the motivations behind serial murders, making this behaviour difficult to assess and authoritatively comment on (Kraemer, Lord & Heilbrun, 2004), however we shall look at some of the research and theories proposed that endeavour to explain the motivations behind serial killing.

Instrumental/Sexual motives
A study by Santtila et al. (2008) identified five different motives for serial homicide, three of them instrumental and two of them sexual:
 * 1) Resources (instrumental)
 * 2) Execution (instrumental)
 * 3) Several victims (instrumental)
 * 4) Rape (sexual)
 * 5) Paraphilic (sexual)

Resources refers to material wealth and money, execution and several victims is referring to status gain, revenge or jealousy, rape is referring to sexual intercourse and paraphilic refers to sexual and often violent fantasies (Santtila et al.).

Holmes and DeBurger's four motivations
 Name: Gary_Ridgway Gary Ridgway Nickname: “The Green River Killer” Birth: February 20 1949 Death: in jail serving life imprisonment without parole Number of victims: confessed to 71, estimates up to 90 Method of killing: strangulation Active: 1982-2001

Holmes and DeBurger (1988) grouped the motivation behind serial homicide into four different categories:
 * 1) Visionary – murder as the consequence of delusions or hallucinations instructing the person to kill.
 * 2) Mission-oriented – goal directed murder, usually with an ambition or objective to kill certain types of people, e.g. prostitutes, elderly women, African-Americans, promiscuous girls.
 * 3) Hedonistic – committed for the thrill and pleasure offered by the planning, perpetration and aftermath of the murder.
 * 4) Power control – enjoyment and satisfaction gained from dominating the victim.

Kraemer, Lord and Heilbrun's five motivations
Kraemer, Lord and Heilbrun (2004) looked at both serial homicide and single homicide cases (157 offenders/608 victims made up the serial homicide group, 195 offenders/133 victims made up the single homicide group [in some cases there were multiple offenders hence the discrepancy between offender/victim]) and coded each based on five motives:
 * 1) Sex – only coded as such if there were apparent sexual intentions or evidence of a sexual nature, or the offender confessed to the crime being sexual in nature.
 * 2) Profit.
 * 3) Emotion-based – usually related to anger, the homicide may have been caused by an argument or disagreement.
 * 4) Psychosis – only coded as such when there was medical evidence included in the case file that confirmed the offender was suffering from hallucinations or delusions at the time.
 * 5) Unclear.

In a comparison between the single homicide data set and the serial homicide data set Kraemer et al. (2004) found that serial offenders were more likely to commit sexually motivated crimes, to have a female victim, to kill strangers, to use strangulation or ligature weapons as their method of killing, to sexually assault their victim and to bind their victim; whilst single offenders were highly likely to commit emotion-based crimes, to have a male victim, kill people they knew, and to use a gun as their method of killing.

Organised/Disorganised murders
 Nickname: “The Boston Strangler” Birth: unknown Death: unknown Number of victims: 13 Method of killing: strangulation Active: 1962-1964 Albert DeSalvo was arrested in connection to the murders and confessed to them, but there are doubts as to whether he really did commit them.

Knight (2006), as well many other researchers and even the Federal Bureau of Investigation in America, subscribes to the organised and disorganised dichotomy of motivation in regards to serial killers. This theory of motivation is based on the offender’s personality and makes reference to the personality types described earlier in the chapter – narcissistic, antisocial and psychopathic, as well as paranoid, schizoid, bipolar and dissociative (Knight). The more organised killer displays a higher level of sophistication in the committal of their crime; they are evidence aware, discerning when choosing their victims (picking victims who are more defenceless thus increasing the successful outcome of their crime), and more easily able to avoid being identified or detected during the committal of their crime. They display higher intelligence levels and are more likely to be employed, they are experienced in social situations, they often see their victims as objects who are there to be controlled, and they usually feel great exhilaration from the performance of sadistic and violent behaviour (Brantley & Kosky, 2005). Organised killers are often motivated by perverse fantasies that contribute to their predatory behaviour and sexual/sadistic desires and often engage in a competition with themselves and the law enforcement officers attempting to catch them, this can turn into a “practice makes perfect” mind-set with the offender continuing to kill in a bid to “out do” themselves and develop their killing techniques (Brantley & Kosky). As you could probably guess from reading the above description, serial killers are often classed as organised.

Thrill/Lust motives
Holmes and Holmes (1998) divided serial killers motivations into two separate categories: thrill and lust.
 * 1) Thrill: the process of killing is the main focus when a thrill killer commits a homicide; they take pleasure from dominating and controlling the victim and will often engage in torture, seeking to inflict the most amount of pain and degradation upon their victim before they die. They have been known to revive a victim if they fall unconscious before the thrill killer has completed their cruelty, some thrill killers have stated that they have become infuriated if a victim has died before they planned, but once the victim is dead the thrill killer quickly loses interest.
 * 2) Lust: sex is the main motive behind a lust killer’s desire to kill, they seek sexual gratification through the act of killing and this may not be “conventional” or overtly obvious as by raping or sexual assaulting/molesting the victim, but may include necrophilia, use of an object, mutilation, dismemberment and trophy taking. Lust killers gain sexual pleasure through the committal of the crime but also after the victim is dead through the above mentioned methods, this is what distinguishes them from thrill killers. Lust killers often plan out their crime and they may partake in abducting or stalking their chosen victim.

Sex is the only motivation
Myers, Husted, Safarik and O’Toole (2006) contend that there is only one motivation behind serial killers whose murders display a sexual nature, and that is sexual desire and gratification – “sexual desire must always be regarded as the basis of sadistic inclinations”. Whereas some researchers have proposed some other motivations for serial homicide, such as power/dominance, Myers et al. (2006) state that the exertion of power/dominance over the victim is merely a necessary component to the completion of the crime, and it may serve to heighten the sexual excitement experienced by the offender. They also present evidence that anger and sexual arousal are conflicting physiological states, meaning that anger cannot be the motivation behind crimes that have a sexual component, as it tends to reduce instead of amplify sexual arousal. Though these crimes obviously have an element of anger to them, the criminal population as a whole has a higher baseline for anger than does their non-criminal counterparts (Myers et al.).

"Motiveless" murders
 Name: Ivan Milat Nickname: “The Backpacker Killer” Birth: December 27 1944 Death: in jail serving 7 life sentences Number of victims: 7 Method of killing: various – stabbing, shooting, decapitation, strangulation Active: 1989-1993 (caught in 1994)

Occasionally there are “motiveless” murders, murders that are committed by individuals who are unable to clarify why they performed such violent behaviour (Holcomb & Daniel, 1988). Police officers, lawyers and psychologists will be among the first to tell you though that there is no such thing as a motiveless crime, behaviour is always motivated by something, in the case of so-called motiveless crimes alcohol intoxication, a psychotic episode, severe brain trauma, a metabolic agitation or epileptic seizure can cause aggression that may be acted out against another person (Holcomb & Daniel, 1988). Holcomb and Daniel conducted a study looking at “motiveless” murders, their results showed that a quarter of their participants obtained a psychotic diagnosis and were more likely to be considered as not responsible for their offence when compared to “motivated” crimes. Interestingly, they also found that the motiveless group were more likely to be unable to remember committing the crime or to deny that they committed the crime and they were more likely to have been released from prison within the past six months before committing the murder (Holcomb & Daniel, 1988).

Evolutionary
An evolutionary perspective has been used to attempt to explain the motivation behind serial homicide and its difference to single homicide. Animals (with human beings included in this classification) only kill for three reasons: self-defence, predation and competition (Kraemer, Lord & Heilbrun, 2004; Myers et al., 2006). Single homicides typically have a male perpetrator and a male victim who usually know one another and so can be defined as the removal of competition (Kraemer et al., 2004). Serial homicides on the other hand, generally have a male perpetrator and a female victim and the victim does not know the perpetrator, these crimes are often likened to predatory behaviour displayed by animals due to the forethought and targeting of victims (Kraemer, Lord & Heilbrun, 2004; Myers et al., 2006).

"I like to dissect girls. Did you know I'm utterly insane?"  Name: Henry Lee Lucas Nickname: N/A Birth: August 23 1936 Death: March 13 2001 [died in prison of heart failure] Number of victims: 11, confessed to 600, estimated believably at 350 Method of killing: shooting and stabbing Active: 1960-1983
 * Feeling brave enough to check out some movies inspired by real-life serial killers?

 * Psycho (1960) – based (in part) on Ed Gein
 * The Honeymoon Killers (1970) – based on Martha Beck & Raymond Fernandez [The Lonely Hearts Killers]
 * 10 Rillington Place (1971) – based on John Christie
 * Henry: Portrait of a Serial Killer (1986) – based on Henry Lee Lucas
 * The Silence of the Lambs (1991) – based (in part) on Ed Gein, Ted Bundy and Gary Heidnik
 * Natural Born Killers (1994) – a controversial film that looks at the media’s adoration of serial killers
 * Citizen X (1995) – based on Andrei Chikatilo
 * Se7en (1995)
 * American Psycho (2000)
 * From Hell (2001) – based on Jack the Ripper
 * Dahmer (2002) – based on Jeffery Dahmer
 * Monster (2003) – based on Aileen Wuornos
 * Wolf Creek (2005) – based (in part) on Ivan Milat and the disappearance of Peter Falconio
 * The Hunt for the BTK Killer (2005) – based on Dennis Rader
 * Zodiac (2007) – based on the Zodiac murders

 <div style=""> Name: Kenneth Bianchi and Angelo Buono Nickname: “The Hillside Strangler” Birth: May 22 1951/October 5 1934 Death: life imprisonment. Buono – September 21 2001 [heart attack] Number of victims: 14 Method of killing: strangulation Active: 1977-1978 (caught in 1979)
 * Serial killers talking about the reasons why they killed.

"What I did was not for sexual pleasure. Rather it brought me some peace of mind."
 * Andrei Chikatilo

"The fantasy that accompanies and generates the anticipation that precedes the crime is always more stimulating than the immediate aftermath of the crime itself."
 * Ted Bundy

"I was literally singing to myself on my way home, after the killing. The tension, the desire to kill a woman had built up in such explosive proportions that when I finally pulled the trigger, all the pressures, all the tensions, all the hatred, had just vanished, dissipated, but only for a short time."
 * David Berkowitz

"The women I killed were filth-bastard prostitutes who were littering the streets. I was just cleaning up the place a bit."
 * Peter Sutcliffe

"When this monster entered my brain, I will never know, but it is here to stay. How does one cure himself? I can't stop it, the monster goes on, and hurts me as well as society. Maybe you can stop him. I can't."
 * Dennis Rader

"That is my ambition, to have killed more people - more helpless people - than any man or woman who has ever lived."
 * Jane Toppan

"I had a compulsion to do it."
 * Ed Gein

"I was born with the devil in me. I could not help the fact that I was a murderer, no more than the poet can help the inspiration to sing… I was born with the evil one standing as my sponsor beside the bed where I was ushered into the world, and he has been with me since."
 * H. H. Holmes

"We've all got the power in our hands to kill, but most people are afraid to use it. The ones who aren't afraid, control life itself."
 * Richard Ramirez

"You know, if I wanted to kill somebody, I’d take this book and beat you to death with it. And I wouldn't feel a thing. It’d be just like walking to the drug store."
 * Charles Manson

"I don't remember killing anyone, I could have done it without knowing it. I am not sure if I did it."
 * Kenneth Erskine

"I just liked to kill, I wanted to kill."
 * Ted Bundy

Summary
<div style=""> Name: Andrei Chikatilo Nickname: “The Butcher of Rostov” “The Red Ripper” Birth: October 16 1936 Death: February 14 1994 [executed – gunshot to the head] Number of victims: 53 Method of killing: various – strangulation and stabbing Active: 1970-1990

This chapter focused on violent (criminal) behaviour and some of the motivations behind it. First off we took a quick run through some definitions so that we knew what was meant when we talked about aggression, violence and motivation. Aggression is a feeling of antagonism usually directed against another person (but can be directed against oneself) and can result in an outburst of violent behaviour designed to harm or intimidate another person. It can be termed physical, verbal or psychological. Violence is the use of physical force that is intended to cause pain and injure or kill another person or group of persons, or to damage or destroy property. Violence can be categorised as physical, psychological, sexual or deprivation. Motivation is the psychological process that directs, energises and influences our behaviour and helps us achieve personal goals. We then took a look at what constitutes a violent crime – murder and nonnegligent manslaughter, aggravated assault, forcible rape and robbery, and read through the FBI definitions for the first three. Next we focused on the biological factors that can influence violent behaviour. It has been estimated that at least 50% of the variance in violent behaviour is determined by a person’s genes and with the use of procedures such as brain stimulation, neuronal tracing and lesioning of particular brain areas, a few structures have been revealed to take part in motivating certain forms of violent behaviour - the amygdala and the lateral, medial and dorsal hypothalamus. Neurotransmitters such as serotonin have also been linked to violent behaviour, a number of studies have looked at the correlation between serotonin levels and violence and found that those with low serotonin levels exhibit more violent behaviour. We then moved on to personality and the motivation behind violence, beginning with Farrington’s four stages of the criminal process – wanting something, choosing a method to obtain it, intensification of motivation and committal of the crime. Then we looked at how the personality traits of antisocial personality disorder and narcissistic personality disorder can influence an individual to engage in violent/criminal behaviour, with Coid’s and Gudjonsson and Sigurdsson’s studies as examples of how these personality types can translate into criminal motivation. We moved on to how it is important to identify the motive behind a crime to both understand and attempt to prevent it from happening again, and to secure a conviction against an offender in court. Schlesinger put forward a criminal motivation framework that was comprised of environmental, situational, impulsive, catathymic, and compulsive motivations and divided crimes into one of the above categories depending on their motivation. We then took a more in-depth look at homicide and serial homicide, perhaps the two most extreme manifestations of violent criminal behaviour, especially in the case of serial homicide. Instrumental and expressive killings were defined, instrumental killings are pre-meditated and, as such, are connected to serial murders, expressive killings are more impulsive and “passionate” and so are connected to single murders. We then moved on to look at multiple theories to do with the motivation behind serial killings proposed by Santtila et al., Holmes and DeBurger, Kraemer, Lord and Heilbrun, Knight, Holmes and Holmes, and Myers et al., as well as so called “motiveless” murders. Finally, a quick look at the evolutionary perspective was presented, which contended that killing only ever occurs for three reasons – self-defence, predation and competition.


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User:EGM6321.F10.TEAM1.HW6

Find
Show that Eq1.3 is a valid solution to 1.1 based on Eq1.2, using the variation of parameters method.

Solution
First let's put the Legendre equation into the standarm form.

$${y}''-\frac{2x}{1-x^{2}}{y}'+\frac{2}{1-x^{2}}y=0$$ We are given the first homogenous solution of this differential equation so we know we can use variation of parameters to solve for the second solution.

$$u_{1}=P_{2}(x)=\frac{1}{2}(3x^{2}-1)$$

$$\begin{align} & y=Uu_{1} \\ & {y}'={U}'u_{1}+U{u}'_{1} \\ & {y}={U}u_{1}+2{U}'{u}'_{1}+U{u}''_{1} \\ & {U}u_{1}+2{U}'{u}'_{1}+U{u}_{1}+\frac{2}{x}({U}'u_{1}+U{u}'_{1})+Uu_{1}=0 \\ & U({u}_{1}+\frac{2}{x}{u}'_{1}+u_{1})+{U}'(2{u}'_{1}+\frac{2}{x}u_{1})+{U}u_{1}=0 \\ \end{align}$$ We know that the given solution when plugged into the differential equation will give us zero as seen in the following. Therefore, we can use a reduction of order method and integrate through to solve for the second homogenous solution.

$$\begin{align} & {u}''_{1}+\frac{2}{x}{u}'_{1}+u_{1}=0 \\ & \text{Let }z(x)={U}' \\ & z(2{u}'_{1}-\frac{2x}{1-x^{2}}u_{1})+{z}'u_{1}=0 \\ & 2\int{\frac{{u}'_{1}}{u_{1}}dx}-2\int{\frac{x}{1-x^{2}}dx}+\int{\frac{z}dx}=0 \\ & \ln (u^{2}_{1})+\ln (x^{2}-1)+\ln (z)=C_{1} \\ & z(x)=C_{1}\frac{1}{u^{2}_{1}(x^{2}-1)} \\ & {U}'=z(x) \\ & U=C_{2}+C_{1}\int{\frac{dx}{u^{2}_{1}(x^{2}-1)}} \\ & U=C_{2}+C_{1}\int{\frac{4}{(3x^{2}-1)^{2}(x^{2}-1)}}dx \\ & \text{Used Wolfram Alpha for this integration}\text{.} \\ & \int{\frac{4}{(3x^{2}-1)^{2}(x^{2}-1)}}dx=\frac{1}{2}(\frac{6x}{3x^{2}-1}+\ln \frac{1-x}{1+x}) \\ & U=C_{2}+C_{1}[\frac{1}{2}(\frac{6x}{3x^{2}-1}+\ln \frac{1-x}{1+x})] \\ & y=Uu_{1}=UP_{2} \\ & y=\frac{C_{2}}{2}(3x^{2}-1)+C_{1}[\frac{3}{2}x+\frac{1}{4}(3x^{2}-1)\ln \frac{1-x}{1+x}] \\ & y=\frac{C_{2}}{2}(3x^{2}-1)-C_{1}[\frac{1}{4}(3x^{2}-1)\ln \frac{1+x}{1-x}-\frac{3}{2}x] \\ \end{align}$$

Author Paul Lang

Find
Solve Eq2.1 and Eq2.2. reference King p28, problem 1.1ab

Solution
First let's solve equation 2.1. First let's put the differential equation into the standard form. $${y}''-\frac{x}{x-1}{y}'+\frac{1}{x-1}y=\frac{f(x)}{x-1}=F(x)$$ We assume a trail solution.

$$\ u_{1}=\exp (x)$$

$$\begin{align} & y=Uu_{1} \\ & {y}'={U}'u_{1}+U{u}'_{1} \\ & {y}={U}u_{1}+2{U}'{u}'_{1}+U{u}''_{1} \\ \end{align}$$ Now we plug these 3 equations above back into the original differential equation and rearrange. $$\begin{align} & {U}u_{1}+2{U}'{u}'_{1}+U{u}_{1}-\frac{x}{x-1}({U}'u_{1}+U{u}'_{1})+\frac{1}{x-1}Uu_{1}=F(x) \\ & U({u}_{1}-\frac{x}{x-1}{u}'_{1}+\frac{1}{x-1}u_{1})+{U}'(2{u}'_{1}-\frac{x}{x-1}u_{1})+{U}u_{1}=F(x) \\ \end{align}$$ Because we assumed that our trial solution is indeed a solution of the differential equation it must satisfy the homogenous case.

$$\begin{align} & {u}''_{1}-\frac{x}{x-1}{u}'_{1}+\frac{1}{x-1}u_{1}=0 \\ & \text{Let }z(x)={U}' \\ & z(2{u}'_{1}-\frac{x}{x-1}u_{1})+{z}'u_{1}=F(x) \\ \end{align}$$

$$\begin{align} & \text{Let }a_{0}=(2\frac{{u}'_{1}}{u_{1}}-\frac{x}{x-1}) \\ & \text{Let }A(x)=\frac{F(x)}{u_{1}}=\frac{f(x)}{u_{1}(x-1)} \\ & {z}'+a_{0}z=A(x) \\ \end{align}$$ We solve by using an integrating factor. $$\begin{align} & h(x)=\exp \{\int{a_{0}dx\}} \\ & \int{a_{0}dx}=\int{2\frac{{u}'_{1}}{u_{1}}dx-\int{\frac{x}{x-1}dx}} \\ & h(x)=\exp \{\ln (u{}_{1}^{2})-x-\ln (x-1)\} \\ & h(x)=\exp \{\ln [\exp (2x)]-x-\ln (x-1)\} \\ & \therefore h(x)=\frac{\exp (x)}{x-1} \\ \end{align}$$ Therefore we know the following. $$\begin{align} & z=\frac{x-1}{\exp (x)}(k_{2}+\int{\frac{\exp (x)}{x-1}Adx)} \\ & {U}'=z \\ & U=k_{1}+\int{k_{2}\frac{x-1}{\exp (x)}}dx+\int{\frac{x-1}{\exp (x)}\int{\{\frac{\exp (x)}{x-1}A(x)dx\}dx}} \\ & y=Uu_{1} \\ & \therefore y=k_{1}\exp (x)-k_{2}\frac{\exp (x)}{\exp (x)}x-\frac{\exp (x)}{\exp (x)}x\int{\frac{\exp (x)}{x-1}A(x)dx} \\ \end{align}$$

Now let's solve the differential equation 2.2. Again we start off by putting it into the standard form.

$${y}''+\frac{2}{x}{y}'+y=\frac{f(x)}{x}=F(x)$$ Assume a trial solution and due to foresight of the problem take the derivative of the assumed solution.

$$\begin{align} & u_{1}=\frac{\sin (x)}{x} \\ & {u}'_{1}=\frac{x\cos (x)-\sin (x)}{x^{2}} \\ \end{align}$$

The rest of the solution is completed in the same manor as the above solution.

$$\begin{align} & y=Uu_{1} \\ & {y}'={U}'u_{1}+U{u}'_{1} \\ & {y}={U}u_{1}+2{U}'{u}'_{1}+U{u}''_{1} \\ & {U}u_{1}+2{U}'{u}'_{1}+U{u}_{1}+\frac{2}{x}({U}'u_{1}+U{u}'_{1})+Uu_{1}=F(x) \\ & U({u}_{1}+\frac{2}{x}{u}'_{1}+u_{1})+{U}'(2{u}'_{1}+\frac{2}{x}u_{1})+{U}u_{1}=F(x) \\ & {u}''_{1}+\frac{2}{x}{u}'_{1}+u_{1}=0 \\ & \text{Let }z(x)={U}' \\ & z(2{u}'_{1}+\frac{2}{x}u_{1})+{z}'u_{1}=F(x) \\ & \text{Let }a_{0}=(2\frac{{u}'_{1}}{u_{1}}+\frac{2}{x}) \\ & \text{Let }A(x)=\frac{F(x)}{u_{1}}=\frac{f(x)}{u_{1}x} \\ & {z}'+a_{0}z=A(x) \\ & h(x)=\exp \{\int{a_{0}dx\}} \\ & \int{a_{0}dx}=\int{2\frac{{u}'_{1}}{u_{1}}dx+\int{\frac{2}{x}dx}} \\ & h(x)=\exp \{2\int{\frac{x\cos (x)-\sin (x)}{x^{2}}}\frac{x}{\sin (x)}dx+2\ln (x)\} \\ & h(x)=\exp \{2\int{\frac{\cos (x)}{\sin (x)}dx-2\int{\frac{dx}{x}+\ln (x^{2})}}\} \\ & h(x)=\exp \{2\ln [\sin (x)]\} \\ & \therefore h(x)=\sin ^{2}(x) \\ & z=\frac{1}{\sin ^{2}(x)}\{k_{2}+\int{\sin ^{2}(x)A(x)dx\}} \\ & {U}'=z \\ & U=k_{1}+\int{k_{2}\frac{1}{\sin ^{2}(x)}}dx+\int{\frac{1}{\sin ^{2}(x)}\int{\{\sin ^{2}(x)A(x)dx\}dx}} \\ & \int{\frac{1}{\sin ^{2}(x)}dx=\int{\csc ^{2}(x)dx}}=-\cot (x) \\ & y=Uu_{1} \\ & y=k_{1}\frac{\sin (x)}{x}-k_{2}\frac{\sin (x)}{x}\frac{\cos (x)}{\sin (x)}-\frac{\sin (x)}{x}\frac{\cos (x)}{\sin (x)}\int{\sin ^{2}(x)A(x)dx} \\ & \therefore y=k_{1}\frac{\sin (x)}{x}-k_{2}\frac{\cos (x)}{x}-\frac{\cos (x)}{x}\int{\sin ^{2}(x)A(x)dx} \\ \end{align}$$

Author Paul Lang

Find
Solve Eq3.1 using the direct method. reference King p34 example; also reference Mtgs 29 and 30; see (2) p29-3

Solution
The complementary function is: Knowing: Interchange parameters to obtain $$ P_1(x)=u_1(x) \ $$ and $$ Q_1(x)=u_2(x) \ $$. Next, given the variation of parameters formula: where Obtain particular solution by rewriting Eq3.4 as: Solve for $$ W_0 \ $$ as $$ s \Rightarrow \ 0 \ $$ Given: and Substituting Eq3.7 and Eq3.8 into Eq3.5 results in: As $$ s \Rightarrow \ 0 \ $$ Eq3.9 becomes: Substituting and solving results in: Reducing results in:

Author EGM6321.F10.TEAM1.WILKS 22:47, 13 November 2010 (UTC) EGM6321,F10.TEAM1.WILKS

Find
Solve for and

Solution
Author EGM6321.f10.team1.Kim.TH

Find
1) Solve for in terms of and

2) Find: and identify $$ \left \{h_i \right\} $$ in terms of $$ \left \{\xi _j \right\} $$

3) Find $$ \Delta\ \Psi\ $$ in cylindrical coordinates.

4) Do separation of variables to recover Bessel's function:

Solution
1) find $$ dx_i\ $$

find $$ dx_1 \ $$

find $$ dx_2 \ $$

find $$ dx_3 \ $$ combine $$ dx_i \ $$

identify $$ ds^2 \ $$

find $$ dx_1^2 \ $$

find $$ dx_2 \ $$

find $$ dx_3 \ $$

combine $$ dx_i^2 \ $$

identify $$ h_i \ $$ by 5.7 and 5.29

find $$ \Delta\ \Psi\ $$ for orthogonal coordinates,

solve summation piecewise for $$ xi_1 \ $$

for $$ xi_2 \ $$

for $$ xi_3 \ $$

recombine

4) Bessel's function is given as:

assuming the solution $$ \phi=0 $$

now, taking the laplacian of this:

substitute in 5.46

divide by $$X_1X_2X_3$$

define a variable $$ \alpha $$. Since this is the only term dependent on $$ \xi_3 $$ it must be constant.

substitute in 5.50 and multiply by $$ \xi_1^2 $$

Now, the second term is the only one dependent on $$ \xi_2 $$ so that it can be redefined as a constant term

define variables $$ x=\xi_1\alpha and X_1=y(x)$$

Author Egm6321.f10.team1.allison 00:19, 10 November 2010 (UTC)

Find
Find $$ \Delta\ \Psi\ $$ in spherical coordinates using the mathematical/physics notation as shown in Eq6.1.

Solution
spherical coordinates in math/science notation can be described as:

find $$ dx_i \ $$

find $$ ds^2 \ $$

which can be simplified to:

identify h: from this the h's are determined to be:

find $$ \Delta\ \Psi\ $$ for orthogonal coordinates,

Author Egm6321.f10.team1.allison 23:42, 10 November 2010 (UTC)

Find
1) Verify expression for $$ \Delta\ \Psi\ $$ in elliptical coordinates from wikipedia article, located at: http://en.wikipedia.org/wiki/Elliptic_coordinates. Use the same steps as in problems 6.5 and 6.6. 2) Obtain the separate equations by separation of variables.

Solution
Elliptical coordinates are defined as:

find $$ x_i's $$

find $$ \Delta\ \Psi\ $$ for orthogonal coordinates,

2) find the seperate equations

divide by $$X_1X_2 $$

for this to be true both terms must be constants.

by the same process,

7.25 and 7.26 could then be solved for $$X_1 $$ and $$ X_2 $$ could then be substituted back to solve $$\Psi $$ Author Egm6321.f10.team1.allison 16:57, 17 November 2010 (UTC)

Given
Where $$\ [n/2] $$ = the integer part of $$\ n/2 $$

Find
Verify that Eq8.3 thru Eq8.7 can be written in the form of Eq8.1 or Eq8.2

Solution
Because equations 8.1 and 8.2 are equivalent, it is only necessary to prove that equations 8.3 through 8.7 may be written in the form of equation 8.1.

1) Verify that equation 8.3 may be written in the form of equation 8.1.

Equation 8.3 is given as:

By observing the subscript in equation 8.3, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.10 yields:

After performing the necessary calculations, equation 8.11 becomes:

This expression of equation 8.1 is equivalent to equation 8.3.

Therefore, equation 8.3 may be written as equation 8.1

2) Verify that equation 8.4 may be written in the form of equation 8.1.

Equation 8.4 is given as:

By observing the subscript in equation 8.4, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.15 yields:

After performing the necessary calculations, equation 8.16 becomes:

This expression of equation 8.1 is equivalent to equation 8.4.

Therefore, equation 8.4 may be written as equation 8.1

3) Verify that equation 8.5 may be written in the form of equation 8.1.

Equation 8.5 is given as:

By observing the subscript in equation 8.5, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.20 yields:

After performing the necessary calculations, equation 8.21 becomes:

This expression of equation 8.1 is equivalent to equation 8.5.

Therefore, equation 8.5 may be written as equation 8.1

4) Verify that equation 8.6 may be written in the form of equation 8.1.

Equation 8.6 is given as:

By observing the subscript in equation 8.6, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.25 yields:

After performing the necessary calculations, equation 8.26 becomes:

This expression of equation 8.1 is equivalent to equation 8.6.

Therefore, equation 8.6 may be written as equation 8.1

5) Verify that equation 8.7 may be written in the form of equation 8.1.

Equation 8.7 is given as:

By observing the subscript in equation 8.7, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.30 yields:

After performing the necessary calculations, equation 8.31 becomes:

This expression of equation 8.1 is equivalent to equation 8.7.

Therefore, equation 8.7 may be written as equation 8.1

It has thus been verified that equations 8.3 through 8.7 may be written as such.

Therefore, equations 8.3 through 8.7 may be written in the form of equation 8.1 or 8.2

Author EGM6321.f10.team1.russo 02:40, 12 November 2010 (UTC)

Find
Verify that Eq9.3 thru Eq9.7 are solutions of Legendre Equations as given in Eq9.1 or Eq9.2

Solution
Since equations 9.2 and 9.1 are identical, it is only necessary to prove that equations 9.3 through 9.7 are solutions to equation 9.1. This will be accomplished by taking the first and second derivatives of each potential solution equation, substituting the results into equation 9.1, and then verifying that there exists an integer $$\ n $$ for which the Legendre differential equation is satisfied:

1) Verify that equation 9.3 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.3 yields:

and taking the second derivative of equation 9.3 yields:

Substituting $$\ P_0(x)$$, $$\ P_0'(x)$$, and $$\ P_0''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=0 $$, equation 9.11 becomes:

And when $$\ n=-1 $$ equation 9.11 becomes:

when $$\ n=0 $$ or when $$\ n=-1 $$

2) Verify that equation 9.4 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.4 yields:

and taking the second derivative of equation 9.4 yields:

Substituting $$\ P_1(x)$$, $$\ P_1'(x)$$, and $$\ P_1''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Dividing both sides of equation 9.18 by $$\ x $$ and rearranging the remaining terms yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=1 $$, equation 9.19 becomes:

And when $$\ n=-2 $$ equation 9.19 becomes:

when $$\ n=1 $$ or when $$\ n=-2 $$

3) Verify that equation 9.5 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.5 yields:

and taking the second derivative of equation 9.5 yields:

Substituting $$\ P_2(x)$$, $$\ P_2'(x)$$, and $$\ P_2''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.26 yields:

Dividing both sides of equation 9.27 by $$\ \frac{1}{2}(3x^2 - 1) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=2 $$, equation 9.28 becomes:

And when $$\ n=-3 $$ equation 9.28 becomes:

when $$\ n=2 $$ or when $$\ n=-3 $$

4) Verify that equation 9.6 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.6 yields:

and taking the second derivative of equation 9.6 yields:

Substituting $$\ P_3(x)$$, $$\ P_3'(x)$$, and $$\ P_3''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.35 yields:

Dividing both sides of equation 9.36 by $$\ \frac{1}{2}(5x^3 - 3x) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=3 $$, equation 9.37 becomes:

And when $$\ n=-4 $$ equation 9.37 becomes:

when $$\ n=3 $$ or when $$\ n=-4 $$

4) Verify that equation 9.7 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.7 yields:

and taking the second derivative of equation 9.7 yields:

Substituting $$\ P_4(x)$$, $$\ P_4'(x)$$, and $$\ P_4''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.44 yields:

Dividing both sides of equation 9.45 by $$\ \left(\frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}\right) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=4 $$, equation 9.46 becomes:

And when $$\ n=-5 $$ equation 9.46 becomes:

when $$\ n=4 $$ or when $$\ n=-5 $$

It has thus been verified that equations 9.3 through 9.7 are solutions to equation 9.1.

Therefore, equations 9.3 through 9.7 are valid solutions to Legendre's differential equation as shown in equation 9.1

Author EGM6321.f10.team1.russo 23:08, 11 November 2010 (UTC)

Signatures
Solved problem 1 -- EGM6321.F10.TEAM1.Lang.P 70.185.113.180 11:47, 17 November 2010 (UTC)

Solved problem 2 -- EGM6321.F10.TEAM1.Lang.P 70.185.113.180 11:47, 17 November 2010 (UTC)

Solved problem 3 -- EGM6321.F10.TEAM1.WILKS 22:47, 13 November 2010 (UTC)EGM6321,F10.TEAM1.WILKS

Solved problem 4 -- EGM6321.f10.team1.Kim.TH

Solved problem 5 -- Egm6321.f10.team1.allison 16:58, 17 November 2010 (UTC)

Solved problem 6 -- Egm6321.f10.team1.allison 16:58, 17 November 2010 (UTC)

Solved problem 7 -- Egm6321.f10.team1.allison 16:58, 17 November 2010 (UTC)

Solved problem 8 -- EGM6321.f10.team1.russo 02:55, 12 November 2010 (UTC)

Solved problem 9 -- EGM6321.f10.team1.russo 23:09, 11 November 2010 (UTC)

title: User:EGM6321.F10.TEAM1.HW6


text:
User:Volker.haas/RoundBoxBottom

 * }

title: User:Volker.haas/RoundBoxBottom


text:
User talk:YouthWork

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title: User talk:YouthWork