text:
User:Egm6321.f09.Team1.andy/HW2

= Problem 1: Integrating Factor Method =

Given
General form of a First Order Nonlinear Ordinary Differential Equations(N1-ODE) is given by,

 * {| style="width:70%" border="0" align="center"

F(x, y(x),\dot y(x)) = 0. $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

 * }
 * }

where, $$\displaystyle \dot y(x) = \frac{dy}{dx} $$

Find
Derive the Euler's Integrating factor $$\displaystyle h(x, y)$$, when it is just a function of y, i.e., $$\displaystyle h_x = 0. $$

Solution
As per the exactness condition (1) for a N1-ODE, Equation (1) must be in the form,

 * {| style=";width:70%" border="0" align="center"

M(x, y(x)) + \left[N(x,y(x)) \right] \frac{d}{dx} y(x) = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)

 * }
 * }

and as per the exactness condition (2) the following equation must be satisfied,

 * {| style="width:70%" border="0" align="center"

\Rightarrow \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$
 * $$\displaystyle

\Rightarrow M_y = N_x $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (3)

 * }
 * }

 Euler's Integrating Factors

If, for a given N1-ODE [Equation (1)], condition (1) of Exactness is satisfied but not the condition (2), then we use Euler's method to find an Integrating Factor $$\displaystyle h(x,y)$$ with a constraint that the condition (2) is satisfied after multiplying $$\displaystyle h(x,y)$$ to the 1st order nonlinear ODE.

 * {| style=";width:70%" border="0" align="center"

M(x, y(x)) + \left[N(x,y(x)) \right] \frac{d}{dx} y(x) = 0 $$
 * $$\displaystyle

 * }
 * }

Multiplying throughout by $$\displaystyle dx $$,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow M dx + N dy = 0 $$
 * $$\displaystyle

 * }
 * }

Multiplying Euler's Integrating Factor $$\displaystyle h(x,y) $$, we get,

 * {| style=";width:70%" border="0" align="center"

 * $$\displaystyle

\Rightarrow \underbrace{(hM)}_\bar{M} dx + \underbrace{(hN)}_\bar{N} dy = 0

$$

 * }
 * }

We need to find the Euler Integrating Factor $$\displaystyle h(x,y) $$ such that,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow \bar{M}_y = \bar{N}_x $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (4)

 * }
 * }

which is nothing but,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow (hM)_y = (hN)_x $$
 * $$\displaystyle
 * }
 * }

We know,

 * {| style=";width:70%" border="0" align="center"

\begin{align} \bar{M}_y & = (hM)_y \\ & \Rightarrow h_y M + h M_y \end{align} $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (5)

 * }
 * }

and,

 * {| style=";width:70%" border="0" align="center"

\begin{align} \bar{N}_x & = (hN)_x \\ & \Rightarrow h_x N + h N_x \end{align} $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (6)

 * }
 * }

Substituting Equations (5) and (6) into Equation (4),

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h_x N - h_y M + h (N_x - M_y) = 0 $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (7)

 * }
 * }

If $$\displaystyle h_x = 0$$, then Equation (7) becomes,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow - h_y M + h (N_x - M_y) = 0 $$
 * $$\displaystyle

 * }
 * }

Dividing throughout by h and re-arranging gives,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow \frac{h_y}{h} = \frac{1}{M}(N_x - M_y) $$
 * $$\displaystyle

 * }
 * }

Integrating throughout with respect to y,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow \int\frac{h_y}{h}dy = \int\frac{1}{M}(N_x - M_y) dy $$
 * $$\displaystyle

\Rightarrow \underbrace{\int\frac{\frac{\partial h}{\partial y}}{h}dy}_{log_e\ (h)} = \int\frac{1}{M}(N_x - M_y) dy $$
 * $$\displaystyle

 * }
 * }

Raising to exponential on both sides,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h(y) = \exp \left[\int\frac{1}{M}(N_x - M_y)dy\right] $$
 * $$\displaystyle

 * }
 * }

Let us define,

 * {| style=";width:70%" border="0" align="center"

-g(y) := \frac{1}{M}(N_x - M_y) $$
 * $$\displaystyle

 * }
 * }

Then the integrating factor becomes,

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle \Rightarrow h(y) = \exp \left[-\int_{}^{y} g(s)ds\right] $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

= Problem 2: Finding Integrating Factor =

Given
A non homogeneous L1-ODE-VC [First Order Linear ODE with varying coefficients] is given by,

 * {| style=";width:70%" border="0" align="center"

y' + \frac{1}{x} y = x^2 $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (8)

 * }
 * }

and the integrating factor is given by,

 * {| style=";width:70%" border="0" align="center"

h(x) = \exp \left[ \int_{}^{x} a_0(s)ds \right] $$
 * $$\displaystyle

 * }
 * }

where $$\displaystyle a_0(x) $$ is given by,

 * {| style=";width:70%" border="0" align="center"

a_1(x)y' + a_0(x)y = b(x) $$
 * $$\displaystyle

$$
 * $$\displaystyle \longrightarrow (9)

 * }
 * }

Find
Show that,

 * {| style=";width:70%" border="0" align="center"

h(x) = x $$
 * $$\displaystyle

 * }
 * }

Solution
Given L1-ODE-VC,

 * {| style=";width:70%" border="0" align="center"

y' + \underbrace{\frac{1}{x}}_{a_0(x)} y = x^2 $$
 * $$\displaystyle

 * }
 * }

which means,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h(x) = \exp \left[ \int_{}^{x} \frac{1}{s}ds \right] $$
 * $$\displaystyle

\Rightarrow h(x) = \exp \left[ log_e[x] \right] $$
 * $$\displaystyle

 * }
 * }

Taking natural logarithm on both sides,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow log_e [h(x)] = log_e [e]^{ log_e[x] } $$
 * $$\displaystyle

\Rightarrow log_e [h(x)] = log_e [x] \underbrace{log_e [e]}_{=1} $$
 * $$\displaystyle

 * }

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle \Rightarrow h(x) = x $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |

 * }
 * }

= Problem 3: Finding solution to L1-ODE-VC =

Given
After multiplying the integrating factor $$\displaystyle h(x) = x $$ to the non homogeneous L1-ODE-VC of Equation (8), we get

 * {| style=";width:70%" border="0" align="center"

xy' + y = x^3 $$
 * $$\displaystyle

 * }
 * }

Find
Show that,

 * {| style=";width:70%" border="0" align="center"

y = \frac{x^3}{4} + \frac{c}{x} $$
 * $$\displaystyle

 * }
 * }

where $$\displaystyle c $$ is a constant.

Solution
Solution to a L1-ODE-VC of Equation (9) with integrating factor $$\displaystyle h(x) $$ is given by,

 * {| style=";width:70%" border="0" align="center"

y(x) = \frac{1}{h(x)} \int_{}^{x} h(s)b(s)ds $$
 * $$\displaystyle

 * }
 * }

So, with $$\displaystyle b(x) = x^2 $$ and $$\displaystyle h(x) = x $$, the solution is given by,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow y(x) = \frac{1}{x} \int_{}^{x} s^3 ds $$
 * $$\displaystyle

 * }
 * }

After integration,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow y(x) = \frac{1}{x} \left[\frac{x^4}{4} + c \right] $$
 * $$\displaystyle

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle \Rightarrow y(x) = \frac{x^3}{4} + \frac{c}{x} $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

= Problem 4: To show a L1-ODE-VC is Exact=

Given
A L1-ODE-VC is given as,

 * {| style=";width:70%" border="0" align="center"

\frac{1}{2}x^2y' + \left[x^4 + 10 \right] = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (10)

 * }
 * }

Find
To show that equation (10) is exact.

Solution
Equation (10) satisfies the exactness condition (1), where

 * {| style=";width:70%" border="0" align="center"

M(x,y) = x^4y + 10 $$
 * $$\displaystyle

 * }
 * }

and

 * {| style=";width:70%" border="0" align="center"

N(x,y) = \frac{1}{2}x^2 $$
 * $$\displaystyle

 * }
 * }

The exactness condition (2) is not satisfied because,

 * {| style=";width:70%" border="0" align="center"

N_x = x \ne M_y = x^4 $$
 * $$\displaystyle

 * }
 * }

We'll try to find an integrating factor $$\displaystyle h(x) $$ such that, [please refer Case: 1]

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h = \exp \left[-\int\frac{1}{N}(N_x - M_y)dx \right] $$
 * $$\displaystyle

 * }
 * }

and we know $$\displaystyle N,\ N_x\ and\ M_y\ are\ all\ functions\ x.$$

On substituting,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h = \exp \left[-\int\frac{1}{\frac{x^2}{2}}(x - x^4)dx \right] $$
 * $$\displaystyle

\Rightarrow h = \exp \left[-2 \int \left(\frac{1}{x} - x^2 \right)dx \right] $$
 * $$\displaystyle

\Rightarrow h = \exp \left[\frac{2x^3}{3} - 2 log_e(x)\right] $$
 * $$\displaystyle

 * }

This shows that Equation (11) is  exactly integrable  and so, is exact.

= Problem 5: To show a N1-ODE is Exact=

Given
A N1-ODE is given as,

 * {| style=";width:70%" border="0" align="center"

\frac{y^5}{5}(5x^3+2) + \frac{x^3}{3}y^4y' = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (11)

 * }
 * }

Find
To show that equation (11) is exact.

Solution
Equation (11) satisfies the exactness condition (1), where

 * {| style=";width:70%" border="0" align="center"

M(x,y) = \frac{y^5}{5}(5x^3+2) $$
 * $$\displaystyle

 * }
 * }

and

 * {| style=";width:70%" border="0" align="center"

N(x,y) = \frac{x^3}{3}y^4 $$
 * $$\displaystyle

 * }
 * }

The exactness condition (2) is not satisfied because,

 * {| style=";width:70%" border="0" align="center"

N_x = x^2y^4 \ne M_y = y^4(5x^3+2) $$
 * $$\displaystyle

 * }
 * }

We'll try to find an integrating factor $$\displaystyle h(x) $$ such that, [please refer Case: 1]

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h = \exp \left[-\int\frac{1}{N}(N_x - M_y)dx \right] $$
 * $$\displaystyle

 * }
 * }

On substituting,

 * {| style=";width:70%" border="0" align="center"

\Rightarrow h = \exp \left[-\int\frac{1}{\frac{x^3}{3}y^4}\left(x^2y^4 - y^4(5x^3+2)\right)dx \right] $$
 * $$\displaystyle

\Rightarrow h = \exp \left[-\int\frac{3}{x^3y^4}y^4\left(x^2 -(5x^3+2)\right)dx \right] $$
 * $$\displaystyle

\Rightarrow h = \exp \left[\int(-\frac{3}{x}+15+\frac{6}{x^3})dx \right] $$
 * $$\displaystyle

\Rightarrow h = \exp \left[-3log_ex+15x-\frac{12}{x^2}\right] $$
 * $$\displaystyle

 * }

This shows that Equation (11) is  exactly integrable  and so, is exact.

= Problem 6: To show a N2-ODE is Exact=

Given
A N2-ODE is given as,

 * {| style=";width:70%" border="0" align="center"

(xy)y'' + x(y')^2 + yy' = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (12)

 * }
 * }

and is in the form of

 * {| style=";width:70%" border="0" align="center"

f(x,y,y')y'' + g(x,y,y') = 0 $$
 * $$\displaystyle

 * }
 * }

where,

 * {| style=";width:70%" border="0" align="center"

f(x,y,y') = xy $$
 * $$\displaystyle

 * }
 * }

and

 * {| style=";width:70%" border="0" align="center"

g(x,y,y') = x(y')^2 + yy' $$
 * $$\displaystyle

 * }
 * }

satisfying the exactness condition (1) for N2-ODE.

Find
To show that equation (12) satisfies the exactness conditions (2) and so, is exact.

Solution
The exactness conditions (2) are given by,

 * {| style=";width:70%" border="0" align="center"

f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xp} + pg_{yp} - g_{y} $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (13)

 * }
 * }

and

 * {| style=";width:70%" border="0" align="center"

f_{xp} + pf_{yp} + 2f_{y} = g_{pp} $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (14)

 * }
 * }

where $$\displaystyle p := y' $$

Calculating the terms on the Left Hand Side of the Equation (13),

 * {| style=";width:70%" border="0" align="center"

f_x = y \ \Rightarrow f_{xx} = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (15)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

f_x = y \ \Rightarrow f_{xy} = 1 \ \Rightarrow 2pf_{xy} = 2p $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (16)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

f_y = x \ \Rightarrow f_{yy} = 0 \ \Rightarrow p^2f_{yy} = 0 $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (17)

 * }
 * }

Adding up Equations (15), (16) and (17) to get the Left Hand Side of Equation (13),

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle f_{xx} + 2pf_{xy} + p^2f_{yy} = 2p $$ $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle \longrightarrow (18)
 * }
 * }

Calculating the terms on the Right Hand Side of the Equation (13),

 * {| style=";width:70%" border="0" align="center"

g_x = (y')^2 = p^2\ \Rightarrow g_{xp} = 2p $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (19)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

g_y = y' = p\ \Rightarrow g_{yp} = 1 \ \Rightarrow pg_{yp} = p $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (20)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

-g_y = -p $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (21)

 * }
 * }

Adding up Equations (19), (20) and (21) to get the Right Hand Side of Equation (13),

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle g_{xp}+pg_{yp}-g_y = 2p $$ $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle \longrightarrow (22)

 * }
 * }

Hence, Equation (18) and Equation (22) are equal and so, Equation (13) is satisfied.

Calculating the terms on the Left Hand Side of the Equation (14),

 * {| style=";width:70%" border="0" align="center"

f_x = y \ \Rightarrow f_{xp} = 0 $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (23)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

f_y = x \ \Rightarrow f_{yp} = 0 \ \Rightarrow pf_{yp} = 0 $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (24)

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

f_y = x \ \Rightarrow 2f_{y} = 2x $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (25)

 * }
 * }

Adding up Equations (23), (24) and (25) to get the Left Hand Side of Equation (14),

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle f_{xp} + pf_{yp} + 2f_{y} = 2x $$ $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle \longrightarrow (26)

 * }
 * }

Right Hand Side of Equation (14),

 * {| style=";width:70%" border="0" align="center"

g_{p} = 2xp $$
 * $$\displaystyle

 * }
 * }

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle g_{pp} = 2x $$ $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |
 *
$$\displaystyle \longrightarrow (27)

 * }
 * }

Hence, Equations (27) and (26) are equal, and so, Equation (14) is satified

This shows that Equation (12), which is a N2-ODE, is  exact .

= Problem 7: Second condition of exactness conditions (2) for a N2-ODE =

Given
A general N2-ODE is given as,

 * {| style=";width:70%" border="0" align="center"

F(x,y,y',y'') = 0 $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (28)

 * }
 * }

and satisfies exactness condition (1) if it is in the form,

 * {| style=";width:70%" border="0" align="center"

f(x,y,p)y'' + g(x,y,p) = 0 $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (29)

 * }
 * }

where $$\displaystyle p:=y' $$

Find
To derive 2nd condition of exactness condtions(2) of a N2-ODE, which is given by

 * {| style=";width:70%" border="0" align="center"

f_{xp} + pf_{yp} + 2f_{y} = g_{pp} $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (30)

 * }
 * }

Solution
A N2-ODE of Equation (28) is exact, if there exists a first integral $$\displaystyle \phi(x,y,p) $$ such that,

 * {| style=";width:70%" border="0" align="center"

F(x,y,y',y'') = \frac{d}{dx} \phi(x,y,p) $$
 * $$\displaystyle

 * }
 * }

which is given by,

 * {| style=";width:70%" border="0" align="center"

F = \phi_x + \phi_y y' + \phi_p y'' $$
 * $$\displaystyle

 * }
 * }

Rearranging,

 * {| style=";width:70%" border="0" align="center"

F = \phi_p y''+ (\phi_x + \phi_y p) $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (31)

 * }
 * }

Comparing Equations (29) and (31),

 * {| style=";width:70%" border="0" align="center"

f(x,y,p) = \phi_p $$
 * $$\displaystyle

 * }
 * }

and

 * {| style=";width:70%" border="0" align="center"

g(x,y,p) = \phi_x + \phi_y p $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (32)

 * }
 * }

Differentiating Equation (32) partially with respect to p,

 * {| style=";width:70%" border="0" align="center"

g_p = \phi_{xp} + (\phi_{yp})p + \phi_y $$
 * $$\displaystyle

 * }
 * }

We know,

 * {| style=";width:70%" border="0" align="center"

\begin{align} \phi_{xp} & = \phi_{px} \\ & = (\phi_p)_x \\ & = f_x \end{align} $$
 * $$\displaystyle

 * }
 * }

and,

 * {| style=";width:70%" border="0" align="center"

\begin{align} \phi_{yp} & = \phi_{py} \\ & = (\phi_p)_y \\ & = f_y \end{align} $$
 * $$\displaystyle

 * }
 * }

and so,

 * {| style=";width:70%" border="0" align="center"

g_p = f_x + pf_y + \phi_y $$ $$
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (33)

 * }
 * }

Again, differentiating the Equation (33) partially with respect to p,

 * {| style=";width:70%" border="0" align="center"

g_{pp} = f_{xp} + pf_{yp} + f_y+ \phi_{yp} $$
 * $$\displaystyle

 * }
 * }

Substituting $$\displaystyle \phi_{yp} = f_y $$, we get

 * {| style=";width:70%" border="0" align="center"

$$\displaystyle g_{pp} = f_{xp} + pf_{yp} + 2f_y $$
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style= |

 * }
 * }

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University of Florida/Egm6321/f09.Team2/HW2

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Problem #1
Problem Statement: Find the Euler Integrating Factor h(y) for a non-linear 1st order ODE for the particular case where we assume hxN=0

Two conditions must be satisfied in order for a nonlinear 1st order ODE to be exact.

The first condition is that $$F(x,y,y')=0$$ must be in the form $$M(x,y)+N(x,y)y'=0$$

The second condition is that $$M_y=N_x$$. This example will show how to satisfy the second condition.

Start with equation $$M(x,y)+N(x,y)y'=0$$ which already satisfies the first condition.

If we assume the equation is not exact we can multiply it by an unkown factor $$h(x,y)$$ (Euler Integrating Factor) to attempt to find a factor that will make the equation exact.

The equation becomes:

$$h(x,y)[M(x,y)dx+N(x,y)dy]=0$$

$$(hM)dx+(hN)dy=0, where\ hM=\overline M, and\ hN=\overline N$$

$$\overline M_y=(hM)_y=h_yM+hM_y$$

$$\overline N_x=(hN)_y=h_xN+hN_x$$

Set $$\overline M_y=\overline N_x$$ and rearrange,

$$h_xN-h_yM+h(N_x-M_y)=0$$

To solve for $$h_y$$ we will let $$h_xN=0$$ which implies that $$h_x=0$$ since $$\scriptstyle M\ne 0$$

$$-h_yM+h(N_x-M_y)=0$$, rearrage terms

$${h_y\over h}dy={1\over M}(N_x-M_y)$$ take the integral of both sides,

$$\int {h_y\over h}dy=\int {1\over M}(N_x-M_y)dy$$

If the right hand side of the integral is only a function of y then it can be written as $$g(y)$$

$${1\over M}(N_x-M_y)=g(y)$$

$$ln\mid h\mid=\int g(y)dy$$

$$h(y)=exp\int_{}^{y}g(s)ds$$  You have lost a negative sign in your expression for $$g(y)$$--Egm6321.f09.TA 02:38, 28 September 2009 (UTC)

Problem 2
Problem Statement: Given the Non-homogeneous linear 1st order ODE with VC $$y'+{1\over x}y=x^2$$ show the steps to obtain $$y={x^3\over 4}+{C\over x}$$ where C=Const.

The first step is to check if the given equation is in the proper form:

$$P_{(x)}y'+Q_{(x)}y=R_{(x)}$$ ; Yes [pg.(8-1) Eg.(1)]

If $$P_{(x)}\ne 0$$ for all (x) then:

$$y'+{Q_{(x)}\over P_{(x)}}y={R_{(x)}\over P{(x)}}$$

In this form we can use the Integration Factor Method to solve for y

$$N_{(x,y)}{dy\over dx}+M_{(x,y)}=0$$

Multiply by $$h_{(x,y)}$$

$$h_{(x,y)}[N_{(x,y)}dy+M_{(x,y)}dx]$$

$$(hM)dx+(hN)dy=0$$ where $$(hM)=\overline M;\ (hN)=\overline N$$

Satisfy the exactness condition $$\overline M_y=\overline N_x$$ where:

$$\overline M_y=(hM)_y=h_yM+hM_y$$

$$\overline N_x=(hN)_x=h_xN+hN_x$$

$$h_xN-h_yM+h(Nx-My)=0$$

To solve for h(x,y) we must make an assumption that h is a function of (x) only.  You do not  have to make this assumption. You have chosen $$h(x,y)=h(x)$$ instead of $$h(x,y)=h(y)$$. --Egm6321.f09.TA 02:42, 28 September 2009 (UTC)

Assume hyM=0, so our equation becomes;

$$h_xN+h(N_x-M_y)=0$$

$${h_x\over h}=-{1\over N_{(x,y)}}(N_x-M_y)$$

Let $$N_{(x,y)}=1$$ and $$M_{(x,y)}={Q_{(x)}\over P_{(x)}}y={1\over x}y$$

$$N_x=0$$ and $$M_y={1\over x}$$

Substitute back into the right hand side of the equationto verify h is a function of (x) only,

$${1\over 1}(0-{1\over x})={1\over x}=f(x)$$

Using that result we can now solve for $$h$$

$$\int {h_x\over h}dx=\int {1\over x}dx$$

$$ln|h|=ln|x|$$

$$h_x=x$$

Now that we have $$h_x$$ we can multiply through our original equation to obtain,

$$(x)y'+(x){1\over x}y=(x)x^2\ =xy'+y=x^3$$

Integrating yields:

$$xy={x^4\over 4}+C$$ divide by x,  Show the steps - the terms must be grouped together as the derivative of a product before you can integrate this. It is not directly integrable in this form.--Egm6321.f09.TA 02:42, 28 September 2009 (UTC)  $$y={x^3\over 4}+{C\over x}$$

Problem #3
Problem Statement: Show that $$ \frac{1}{2}x^{2}y'+[x^{4}y+10]= 0 $$  is exact.

In this problem, we are asking if: (1) this equation is only a function of x, and (2) it is "exact", meaning exact or exactly integrable by the Integrating Factors Method.

To be exact, it must meet 2 conditions:

(1)it must meet the form $$M(x,y) + N(x,y)y' = 0$$ (2)$$M_y = N_x$$

If $$M(x,y)dy = a(x)$$, then $$M(x,y) = a(x)y + k$$ where k is a constant.

Similarly: $$N(x,y)dx = b(x)$$, then $$N(x,y) = \int_{}^{x}b(s)\, ds = c(x) = \overline{b}(x)$$. Replacing M and N in our initial form

$$M(x,y)dx + N(x,y)dy = [a(x)y + k]dx + c(x)dy = 0$$

Dividing by $$dx$$ and re-arranging: $$[a(x)y +k] + \overline{b}(x) y' = 0$$. Our initial problem meets condition 1, where $$\overline{b}(x) = \frac{1}{2}x^2 = N(x,y)$$ $$a(x) + k = x^4y + 10 = M(x,y)$$ where $$k = 10$$

For condition 2, $$M_y = \frac{d}{dy}[a(x) + k] = \frac{d}{dy}(x^4y + 10) = x^4$$ and $$N_x = \frac{d}{dx}(\frac{1}{2}x^2) = x$$

But these two are not equal. Therefore, using the Integrating Factors Method, we must find $$h(x,y)$$ such that $$ (hM)_y = (hN)_x$$ and thereby satisfy condition 2. Differentiating and rearranging terms gives us: $$h_xN - h_yM + h(N_x - M_y) = 0$$. If we assume $$h_y=0$$ since $$M_y$$ and $$N_x$$ are both functions of x, and rearrange terms, we get: $$\frac{h_x}{h} = \frac{-1}{N}(N_x - M_y)dx$$. Integrating both sides creates $$log (h)$$. Solving for h: $$ h = e$$^$$(-2)(\int_{}^{x}N_x - M_y,\ dx$$. We know $$N_x$$ and $$M_y$$.  Therefore $$h = e$$^$$(-2)(\frac{1}{2}x^2 - \frac{1}{5}x^5)$$. Multiplying $$M(x,y)$$ and $$N(x,y)$$ by $$h$$ causes our function to meet the 2nd condition and therefore be exact.

$$(e$$^$$(\frac{2}{5}x^5 - \frac{1}{2}x^2) (\frac{1}{2}x^{2}y'+[x^{4}y+10]= 0 $$ is exact.  This equation is incomplete. There are parenthesis missing. Where are the differentials?--Egm6321.f09.TA 02:47, 28 September 2009 (UTC)  You need to show that you have arrived at the correct integrating factor. Specifically: $$\frac{\partial\overline{M}}{\partial y}=\frac{\partial\overline{N}}{\partial x}$$.--Egm6321.f09.TA 02:47, 28 September 2009 (UTC)

Problem #4
Problem Statement: Show that $$ (\frac{1}{3}x^{3})(y^{4})y'+(5x^{3}+2)(\frac{1}{5}y^{5})=0$$ is an exact nonlinear, first order ODE.

To prove this equation is an exact, non-linear, 1st order ODE, it must meet 2 conditions: (1) it must fit the form $$M(x,y)y' + N(x,y) = 0$$ and (2) $$M_y = N_x$$

It is obvious from simple observation that our equation meets condition 1; however it fails to meet condition 2. $$M(x,y) = \frac{1}{3}x^3y^4$$  and   $$N(x,y) = (5x^3 - 2)(\frac{1}{5}y^5)$$  then $$M_y = \frac{4}{3}x^3y^3$$ and $$N_x = 5x^3y^4$$. Clearly $$M_y \ne N_x$$.

To meet condition 2, we must find h(x,y) such that $$(hM)_y = (hN)_x$$. As $$M_y$$ and $$N_x$$ only differ in $$y$$, let us assume that $$h_x = 0$$. Differentiating and rearranging terms gives us: $$\frac{h_y}{h} = \frac{1}{M}(N_x - M_y)dx$$. Integrating both sides creates $$log h$$. Solving for h: $$ h = e$$^$$(\frac{1}{M}\int_{}^{y}N_x - M_y,\ dy)$$. We know $$N_x$$ and $$M_y$$. Substituting for $$N_x$$ and $$M_y$$ and simplifying: $$h = e$$^$$(15y-4(log(y))$$. This, multiplied by our original equation will cause it to be an exact, nonlinear, first order ODE.  The $$\frac{1}{M}$$ should be inside the integral.--Egm6321.f09.TA 02:51, 28 September 2009 (UTC) The simpliest method to prove this equation as an exact, nonlinear, first order ODE is to show that (1) it fits the form for a class of exact, non-linear 1st order ODE's.  Specifically: $$\overline{b}(x)c(y)y' + a(x)\overline{c}(y) = 0$$  where $$\overline{b}(x) = \int_{}^{x} b(x)\, dx$$ and $$\overline{c}(y) = \int_{}^{y} c(y)\, dy$$

If we assume: $$a(x) = 5x^{3}+2$$ $$b(x) = x^{2}$$ and $$c(y) = y^{4}$$ then $$\overline{b}(x) = \int_{}^{x} x^{2}\, dx = \frac{1}{3}x^{3}$$ and $$\overline{c}(y) = \int_{}^{y} y^{4}\, dy = \frac{1}{5}y^{5}$$

Since our equation does fit the form for this class, it is an exact, nonlinear, first order ODE.

 Need to verify that $$\overline{N}_x=\overline{M}_y$$--Egm6321.f09.TA 02:51, 28 September 2009 (UTC)

Problem #5
Problem Statement: Show that the second exactness condition for $$xyy''+x(y')^{2}+yy'=0$$ is satisfied.

The 2nd exactness condition actually consists of 2 parts, and our equation must meet both: (1) fxx + (2p) fxy + (p2) fyy = gxp + p(gyp) - gy

(2) fxp + p (fyp) + 2 (fy) = gpp

where p is y'

In our equation, f(x,y,p) = $$xy$$ and  g(x,y,p) = $$x(y')^{2}+yy'$$

Looking at our equation, for the first half of the condition: fxx = $$\frac{d}{dx}$$[$$\frac{d}{dx}$$($$xy$$)] = $$\frac{d}{dx}$$($$y$$) = 0

fxy = $$\frac{d}{dy}$$[$$\frac{d}{dx}$$($$xy$$)] = $$\frac{d}{dy}$$($$y$$) = 1

fyy = $$\frac{d}{dy}$$[$$\frac{d}{dy}$$($$xy$$)] = $$\frac{d}{dy}$$($$x$$) = 0

gxp = $$\frac{d}{dp}$$[$$\frac{d}{dx}$$($$x(p)^{2}+yp$$)] = $$\frac{d}{dp}$$($$p^{2}$$) = 2p

gyp = $$\frac{d}{dp}$$[$$\frac{d}{dy}$$($$x(p)^{2}+yp$$)] = $$\frac{d}{dp}$$($$p$$) = 1

gy = $$\frac{d}{dy}$$($$x(p)^{2}+yp$$) = p

Plugging back into the first half of our condition, $$0 + 2p(1) + p2(0) = 2p + p(1) - p$$ $$2p = 2p$$  and therefore the first half of the condition is satisfied.

 What is sup? Check to verify that your code has rendered properly.--Egm6321.f09.TA 02:57, 28 September 2009 (UTC)

Looking at the 2nd half of the condition: fxp = $$\frac{d}{dp}$$[$$\frac{d}{dy}$$($$xy$$)] = $$\frac{d}{dp}$$($$y$$) = 0

fyp = $$\frac{d}{dp}$$[$$\frac{d}{dx}$$($$xy$$)] = $$\frac{d}{dp}$$($$x$$) = 0

fy = $$\frac{d}{dy}$$($$xy$$)] = x

gpp = $$\frac{d}{dp}$$[$$\frac{d}{dp}$$($$x(p)^{2}+yp$$)] = $$\frac{d}{dp}$$($$2xp + y$$) = $$2x(p')$$

Plugging back into the second half of our condition (remembering that p = y'), $$0 + p(0) + 2(x)p' = 2x(p')$$ thereby satisfying the second half of the condition.  What is $$p'$$ ? This would be $$ y''$$ and this should not appear in this problem.--Egm6321.f09.TA 02:57, 28 September 2009 (UTC)

Problem #6
Problem Statement: Derive $$f_{xp}+pf_{yp}+2f_{y}=g_{pp}$$ using the following relations:

(1)  $$f(x,y,p):=\phi_{p}(x,y,p)$$ (2)  $$g(x,y,p):=\phi_{x}+\phi_{y}p$$ (3)  $$\phi_{xp}=\phi_{px}$$ (4)  $$\phi_{yp}=\phi_{py}$$

Start by taking the partial derivative of g with respect to p twice: $$g_{pp}=\frac{\partial^2 g}{\partial p^2}$$

$$g_{p}=\phi_{xp}+\phi_{yp}p+\phi_{y}$$ $$g_{pp}=\phi_{xpp}+\phi_{ypp}p+\phi_{yp}+\phi_{yp}$$ $$=>g_{pp}=\phi_{xpp}+\phi_{ypp}p+2\phi_{yp}$$

Now use (1), (3), and (4) to substitute $$\phi_{xpp}$$, $$\phi_{ypp}$$, and $$\phi_{yp}$$

$$f_{x}=\phi_{px}=\phi_{xp}$$ by (1) and (3) $$f_{xp}=\phi_{xpp}$$ $$f_{y}=\phi_{py}=\phi_{yp}$$ by (1) and (4) $$f_{yp}=\phi_{ypp}$$

Therefore,  $$g_{pp}=f_{xp}+f_{yp}p+2f_{y}=f_{xp}+pf_{yp}+2f_{y}$$  Clear and concise.--Egm6321.f09.TA 02:58, 28 September 2009 (UTC)

Problem #7
Problem Statement: Derive $$f_{xx}+2pf_{xy}+p^2f_{yy}=g_{xp}+pg_{yp}-g_{y}$$ using the following relations: (1) $$f(x,y,p):=\phi_{p}(x,y,p)$$ (2) $$g(x,y,p):=\phi_{x}+\phi_{y}p$$ (3) $$\phi_{px}=\phi_{xp}$$ (4) $$\phi_{py}=\phi_{yp}$$ (5) $$\phi_{xy}=\phi_{yx}$$

Solving (1) for $$\phi_{p}$$ and taking the partial with respect to $$x$$ yields $$\phi_{px}=f_{x}$$ Then solving (2) for $$\phi_{x}$$ and taking the partial with respect to $$p$$ yields $$\phi_{xp}=g_{p}-\phi_{yp}p-\phi_{y}$$ Equating these two by (3) and solving for $$\phi_{y}$$ yields $$\phi_{y}=-f_{x}+g_{p}-\phi_{yp}p$$ Performing a similar task, take (1), solve for $$\phi_{p}$$ and take the partial with respect to $$y$$ $$\phi_{py}=f_{y}$$ Then solving (2) for $$\phi_{y}$$ and taking the partial with respect to $$p$$ yields $$\phi_{yp}=\frac{g_{p}}{p}-\frac{g}{p^2}-\frac{\phi_{xp}}{p}+\frac{\phi_{x}}{p^2}$$ Equating these two by (4) and solving for $$\phi_{x}$$ yields $$\phi_{x}=f_{y}p^2-g_{p}p+g+\phi_{xp}p$$

Finally, $$fx=\phi_{px}=\phi_{xp}$$ and $$fy=\phi_{py}=\phi_{yp}$$ as previously shown and so the two equations are (a) $$\phi_{y}=-f_{x}+g_{p}-f_{y}p$$ (b) $$\phi_{x}=f_{y}p^2-g_{p}p+g+f_{x}p$$ Taking the partial with respect to $$x$$ for (a) and the partial with respect to $$y$$ for (b) and equating based on (5) yields $$-f_{xx}+g_{px}-f_{yx}p=f_{yy}p^2-g_{py}p+g_{y}+f_{xy}p$$

Noting $$f_{yx}=f_{xy}$$, $$g_{px}=g_{xp}$$, $$g_{py}=g_{yp}$$, and moving the f terms to the left and the g terms to the right

 $$f_{xx}+2pf_{xy}+p^2f_{yy}=g_{xp}+pg_{yp}-g_{y}$$
 You have performed much unnecessary rearranging of $$\Phi$$ terms to get here. You need to rearrange until the end. Equate $$\Phi_{xy}=\Phi_{yx}$$ using the appropriate relations and your solution will be must more straight forward.--Egm6321.f09.TA 03:01, 28 September 2009 (UTC)

Problem #8
Problem Statement: Equations 4&5 on p.([[media:Egm6321.f09.p10-2.png|10-2]]).

Given:

$$

(8x^{5}y^{'})y^{''} + 2x^{2}y^{'} + 20x^{4}(y^{'})^{2} + 4xy = 0

$$

Show that the Nonlinear 2nd-Order ODE is exact

Solution
Second Condition of Exactness: 10-2 Eq (4),(5)

$$

\ F_{xx} + 2pF_{xy} + p^{2}F_{yy} = G_{xp} + pG_{yp} - G_{y} \

$$

$$

\ F_{xp} + pF_{yp} + 2F_{y} = G_{pp} \

$$

$$

\ F(x,y,p)y^{''} + G(x,y,p) = 0

$$

$$

\ F(x,y,p) = 8x^{5}p \qquad \qquad G(x,y,p) = 2x^{2}p + 20x^{4}p^{2} + 4xy

$$

The Following partial derivatives are then identified:

$$

F_{x}=40x^{4}p  \qquad \qquad  G_{x}= 4xp + 80x^{3}p^{2} + 4y

$$

$$

F_{xx}=160x^{3}p \qquad \qquad  G_{xp}= 4x + 160x^{3}p

$$

$$

F_{xy}=0 \qquad \qquad  G_{y}= 4x

$$

$$

F_{xp}=40x^{4}  \qquad \qquad  G_{yp}= 0

$$

$$

F_{y}=0 \qquad \qquad  G_{p}= 2x^{2} + 40x^{4}p

$$

$$

F_{yy}=0 \qquad \qquad  G_{pp}= 40x^{4}

$$

$$

\ 160x^{3}p +2p(0) +p^{2}(0) = 4x + 160x^{3}p + (0) -4x

\ 160x^{3}p = 160x^{3}p $$

Page 10-2 Eq(4) is then satisfied.

Applying the results in Page 10-2 Eq(5):

$$

\ 40x^{4} + p(0) + 2(0) = 40x^{4}

$$

$$

\ 40x^{4} = 40x^{4}

$$

Page 10-2 Eq(5) is also satisfied. Because Eq(4) and Eq(5) are both satisfied, then the Secound Exactness Condition is satisfied.

Therefore, $$

\ F(x,y,p) = 8x^{5}p \qquad \qquad G(x,y,p) = 2x^{2}p + 20x^{4}p^{2} + 4xy

$$

is Exactness Second ODE is an exact, nonlinear second ODE
 Nice work. One caveat: For your final statement, express the equation in terms of $$x,y,y'$$. The equation involves an unknown function $$y(x)$$, where $$p$$ was a dummy variable which helped us perform differentiations. --Egm6321.f09.TA 03:05, 28 September 2009 (UTC)

Problem #9
Problem Statement: Verify the exactness of the ODE $$\sqrt{x}y''+2xy'+3y=0$$.

Given

$$

\sqrt[]{x}y^{''} + 2xy^{'} + 3y = 0

$$

Prove that the equation is not exact.

Solution
First Condition of Exactness Page10-1 Eq(1):

$$

\ F(x,y,p)y^{''} + G(x,y,p) = 0

$$

$$

\ F(x,y,p) = \sqrt[]{x} \qquad \qquad G(x,y,p) = 2xp + 3y

$$

This satisfy satisfies the First Condition of Exactness. The Second Exactness Condition for a second order ODE is as follows:

$$

\ F_{xx} + 2pF_{xy} + p^{2}F_{yy} = G_{xp} + pG_{yp} - G_{y} \qquad Page 10-2 Eq(4)

$$

$$

\ F_{xp} + pF_{yp} + 2F_{y} = G_{pp} \qquad Page 10-2 Eq(5)

$$

The following partial derivatives are found:

$$

F_{x}= -\frac{1}{2}x^{\frac{-1}{2}}  \qquad \qquad   G_{x}= 2p

$$

$$

F_{xx}= -\frac{1}{4}x^{\frac{-3}{2}}  \qquad \qquad  G_{xp}= 2

$$

$$

F_{xy}= 0 \qquad \qquad \qquad  G_{y}= 3

$$

$$

F_{xp}= 0  \qquad  \qquad \qquad  G_{yp}= 0

$$

$$

F_{y}= 0 \qquad \qquad \qquad  G_{p}= 2x

$$

$$

F_{yy}=0 \qquad \qquad \qquad  G_{pp}= 0

$$

$$

F_{yp}=0 \qquad \qquad

$$

Using these values in Eq(4):

$$

\frac{-1}{4}x^{\frac{-3}{2}} + 2p(0) + p^{2}(0) = 2 + p(0) -3

$$

$$

\frac{-1}{4}x^{\frac{-3}{2}} = -1

$$

Page 10-2 Eq(4) is not satisfied therefore the ODE is not exact.
 Add equation numbers to your solution here so that you can reference the equations in your solution, rather than a transparency that is located somewhere else on the web. This will make your solution self contained and complete in itself.--Egm6321.f09.TA 03:09, 28 September 2009 (UTC)

Page 10-2 Eq(5) can be populated to find:

$$

\ 0 = 0 + p(0) + 2(0)

$$

$$

\ 0 = 0

$$

Eq (5) is satisfied.

In order for the ODE to be exact it must satisfy both Eq(4)and Eq(5), since it fails to do this it is concluded that the ODE is not exact.

Contributing Team Members
Egm6321.f09.Team 2.walker 20:58, 20 September 2009 (UTC) (Walker, Matthew)

Joe Gaddone 14:04, 21 September 2009 (UTC)

Egm6321.f09.Team2.sungsik 19:20, 23 September 2009 (UTC)

Kumanchik 19:53, 23 September 2009 (UTC)


text:
Strategies for Engineered Negligible Senescence

Strategies for Engineered Negligible Senescence is SENS. This area is devoted to learning, teaching, and research related to SENS.

Many educational institutions attempt to cure diseases. There are many teaching hospitals that exist. There are many universities that conduct research into how we can reduce the global disease burden.

Aging is a lot like smoking cigarettes, it is really bad for you.

Discussion questions
🇨🇴
 * Should funding for SENS research be increased? By how much?
 * How can you help to raise awareness for SENS research?
 * How you read the book Ending Aging by Aubrey de Grey? What did you think about it? What did you learn from it?
 * How might SENS eventually help reduce the prevalence of atherosclerosis?

Learning and research resources

 * Fetal tissue transplantation in Regenerative Medicine

Lessons research and discussion

 * Why SENS is ethical and should be adopted as a point of view
 * Controversies surrounding SENS
 * Ethics involving SENS
 * How you can help support SENS research

Wikipedia

 * SENS Research Foundation
 * Strategies for Engineered Negligible Senescence

Papers

 * It is time to classify biological aging as a disease

More

 * Aubrey de Grey's answer to "Is it correct that the WHO has stated that aging is a disease?"


text:
Psychiatric Liberation/Psychopharmaceutical drugs and herbal remedies

SSRIS

Anti-psychotics
 * Atypical
 * Haldol

Herbal
 * Saint Johns Wort


text:
User talk:Eleanor McHugh

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User talk:Kimaule

Hello Former Student!

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Non-free images
Hi! As part of a clean-up of the use of non-free images, I noticed that you used to images under fair use as part of your work at User:Stevenarntson/group spaces/2009summeranalytical/group 4, specifically File:200px-Parental Advisory label.svg.png, File:CCA.gif and File:EwigerJudeFilm.jpg. As the work wasn't in Wikiversity's mainspace, and as it seems that it was for a course that you completed a couple of years ago, I've removed the images and nominated them for deletion. If this is a problem, though, just let me know and I'll do my best to assist. :) - Bilby 08:13, 13 October 2011 (UTC)

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User:Egm6321.f09.Team8

user:egm6321.f09.Team8/HW1

user:egm6321.f09.Team8/HW2

user:egm6321.f09.Team8/HW3

title: User:Egm6321.f09.Team8


text:
User:Egm6321.f09.Team8/HW1

Problem #1 ([[media:Egm6321.f09.p2-2.png|2-2]]): Derive equations (1) and (2) on p.[[media:Egm6321.f09.p1-2.png|1-2]] $$ \frac{d}{dt}f(Y^{1}(t),t)=\frac{\partial f}{\partial s}(Y^{1}(t),t)\dot{Y}^{1}(t)+\frac{\partial f}{\partial t}(Y^{1}(t),t) $$ $$ \frac{d^{2}}{dt^{2}}f=f_{,s}(Y^{1},t)\ddot{Y}^{1}+f_{,ss}(Y^{1},t)(\dot{Y}^{1})+2f_{,st}(Y^{1},t)\dot{Y}^{1}+f_{,tt}(Y^{1},t) $$

Solution:

[[Media:HW1, Page 1.PNG]]

Problem #2 ([[media:Egm6321.f09.p4-2.png|p.4-2]]): Derive Eq.(2) on [[media:Egm6321.f09.p4-2.png|p.4-2]] using int factor. $$ p(x)=Ae^{-x}+x-1 $$

Solution:

[[Media:HW1, Page 2.PNG]]

[[Media:HW1, Page 3.PNG]]

Problem #3 ([[media:Egm6321.f09.p4-3.png|4-3]]): Show that the 1st order ODE is nonlinear. $$M(x,y)=2x^{2}+\sqrt{y} $$ $$N(x,y)=x^{5}y^{3}$$ $$(2x^{2}+\sqrt{y})+x^{5}y^{3}y'=0$$

Solution:

[[Media:HW1, Page 4.PNG]]

Problem #4 ([[media:Egm6321.f09.p5-2.png|5-2]]): Show that $$F(x,y,y')=0$$ in Eq. (3) is a nonlinear 1st order ODE. (Hint: Define the differential operator $$D(.)$$ associated with equation (3).)

Solution:

[[Media:HW1, Page 5.PNG]]

Problem #5 ([[media:Egm6321.f09.p6-1.png|6-1]]): Generating exact nonlinear 1st order ODEs: Let $$ \Phi(x,y)=6x^{4}+2y^{3/2}$$, then $$ M=\Phi_{x}$$ $$ N=\Phi_{y}$$ Complete the details and invent 3 more examples.

Solution:

[[Media:HW1, Page 6.PNG]]

[[Media:HW1, Page 7.PNG]]

title: User:Egm6321.f09.Team8/HW1


text:
User talk:Egm6321.f09.Team8

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title: User talk:Egm6321.f09.Team8


text:
Emerging tech Africa

Overview & background
Introduction to Emerging Technologies is an open online course delivered in French and English, addressing the concerns and opportunities around the use of technologies for learning in Africa.

In early 2009, the African Association of Universities partnered with University of Manitoba to initiate a project (funded by OSIWA): Capacity building on social networking technologies for teaching and learning transformation in West African universities.

The first phase of the project included a multi-day face-to-face workshop in Dakar, Senegal in May (overlapping with Elearning Africa) with educational leaders in West Africa. Slides from Day 2 of this session are available. The second phase of the project is an open online course on Emerging Technologies for Learning.

In order to increase access and impact, learners from around the world are invited to participate in this innovative bi-lingual (English & French) course. Weekly topics will focus on how emerging technologies and open education initiatives can be utilized in the context of Africa. Discussion will centre on the philosophy and ideology of technology, ICT for communication, creation, and content sharing, changing learner/educator roles, and the utilization of learning networks to increase educational opportunities for Africa.

Objectives
* Define digital literacy and detail the impact of emerging technology in information validity, online identity, and privacy.
 * Create artifacts that represent digital literacy skills, including (but not limited to): podcast, video, slidecast, concept map, blog posts, wiki, Second Life avatar, webcast, and PLE image.
 * Evaluate the affordances of technology and utilize a technology selection framework to incorporate new tools into teaching and learning
 * Describe the potential impact on teaching and learning of personal learning environments (PLE) in relation to established learning management systems.
 * Create a graphical representation of your PLE and evaluate its suitability in assisting achievement of personal learning goals.
 * Use online communication tools (Skype, Google Talk, ooVoo) to improve conversation opportunities with learners and peers, both within your institution and globally.
 * Evaluate the impact of emerging technologies on power structures within education systems such as like universities, with particular emphasis on 1) changing roles of academics and learners, and 2) social networked learning theory.
 * Critique existing information cycles (academic publication, public relations, news) and evaluate how emerging technologies and societal trends alter this cycle.
 * Utilize aggregation tools (such as Google Reader and PersonalBrain) in managing personal, professional, and academic information.

These objectives will, throughout the course, be contextualized to the experience of African educators and learners (note: as we developed this course, we were conscious of not making assumptions of the regional and cultural concerns of African participants. As such, we considered skills/attributes of a global nature - i.e. "what does an individual need to know and be able to do to participate in global conversations, mediated by technology?" With this as a starting point, we concluded that the major differences unique to African learners would be related to technology access. The needs for learners from Africa to be familiar with emerging technologies is similar to learners from Canada, China, or Europe. Through out the course, we will rely heavily on learners helping to frame the discussions within the technological constraints unique to Africa). center text

Facilitators
The initial grant application to OSIWA was completed, and face-to-face workshop conducted, by Boubakar Barry, Kathleen Matheos, and George Siemens.

The second phase of the project - this open online course - will be facilitated by George Siemens and Dave Cormier.

How to participate
French version

English version

Course structure
As an open course, learners are encouraged to participate in forums and discussions of their interest and to initiate new learning opportunities as they see necessary. For example, if a learner feels that a sub-group discussing problems around electricity and technology is warranted, we encourage her to set up the site/forum/discussion area. As facilitators, our objective is to provide a suitable structure for innovating thinking and exploration of important ideas.

 * Discussions will be held in various forums:

Moodle (see the French/English links above to the link to different language courses in Moodle)

 * Live discussion sessions:

Regular emails from course facilitators highlighting discussions occurring on blogs and moodle forums. To sign up for emails, please subscribe to this Google Group.

Start date
October 12, 2009

title: Emerging tech Africa


text:
Intro Emerging Tech

English version of Intro to Emerging Tech for Africa

General Information:
Instructor: George Siemens (http://www.elearnspace.org (weekly newsletter on emerging tech) http://www.connectivism.ca)...more info

Contact: gsiemens AT gmail DOT com

Course Tag: AfricaIET09

Course moodle site: Introduction: Emerging Technologies, Africa

Google Group (join to receive regular email updates on course progress and highlighted learner commentary/resources.

Evaluation:
Learners - those funded by the OSIWA grant - will be asked to complete a presentation (for week 8) as well as actively participate in the course activities. Evaluation will be based on the following:
 * 1) Regular, weekly, participation in moodle forum and blog discussions (30%). Forum/blog contributions are important as they provide learners with an opportunity to form connections with each other. These discussions are developmental (i.e. when done on a weekly basis, they provide value in sharing ideas and receiving feedback, rather than when done at one moment toward the end of the course when feedback from others does not contribute to ongoing development). A forum or blog post should be substantive - advancing a thought expressed by articles or by a previous learner's comments, critically evaluating concepts in the course, or applying weekly resources/tools in the context of Africa. Length is secondary to offering insight on the topic.
 * 2) Weekly experimentation with emerging technologies, resulting in an digital artifact shared with others in the course (i.e. blog, podcast, wiki, image, etc.) (30%)
 * 3) Final presentation (40%). More information on your final presentation: http://ltc.umanitoba.ca/wiki/Presentation

Week 1: Considering Literacy in a Digital Age
http://web.archive.org/web/20060206030651/http://www.geocities.com/c.lankshear/Oslo.pdf Digital literacies particularly targeted.

The word 'literacies' has many different meanings in different contexts. With the help of the readings from this week we are trying to expose more complex interpretations of literacies than is sometimes used in order to explore the widest range of possible contributors to success or difficulty inside of an online setting.

 * 1) What literacies do you see as fundamental to a positive online experience?
 * 2) Which ones do you currently possess?
 * 3) Which ones do you think you will need to learn?

This course will argue that, among other things, knowledge is created in the connections between ideas. Your primary responsibility during this course is to engage in discussion with your fellow learners. In this exercise this can be done by replying to comments in the discussion threads. Responses in here are NOT to be directed to the instructors of this course but rather to all learners.

Activities for week 1:

 * 1) Listen to week 1 podcast introduction to the course
 * 2) Post your introduction in the course moodle site
 * 3) After reading the article on literacies, comment on how literacies are reflected in the needs of your university/students/country.

Week 2: Identity & Privacy: Tools of Expression and Presence
In week one, we discussed digital literacies. The ability to participate in the conversations of an era, through the era's prominent technologies/media, requires a focus of our identity. Our identity online is shaped by our participation. Each forum we contribute to, each blog post we author, each comment we post, and each profile we create contributes to our identity. Google (or summize for twitter and technorati for blogs) provides easy access to other who wish to observe or explore our activities.

While there is a bit of overlap between presence and expression tools, distinct elements can be noticed in each.

Tools of presence allow individuals to declare "what they are doing". This is noticed in Skype (the online status indicator), Twitter, and instant messaging. These tools are effective in letting others know where you are and how accessible you are.

Tools of expression allow individuals to declare "who they are". Profiles in Google, Facebook, and other social networking services are used to present ourselves to others. By making our interests (and identity) explicit, others are able to "discover" us. Transparency and discoverability are critical foundations for forming social and learning networks.

Readings

Digital Footprints (.pdf)

Activities:

After reading the "Digital Footprints" article, reflect on ways in which different tools influence your identity online. In particular, comment on concerns that learners face in your country or institution. Most discussions of privacy and identity are focused on west-world views. What types of privacy issues do you and your learners face?

Assignments:

Mid-way through the course, you will be asked to complete a presentation for other learners in the course (see the assessments section of this outline). To begin preparing for your presentation, review the tools listed in the Emerging Technology Handbook. Which tools interest you? Why?

Week 3: Personal Learning Environments
We also need to keep track of the learning that we are doing. Something that you learn from this course might only end up being useful 6 months or 6 years from now. Having a clear way of tracking the pieces of work that you are doing is critical to life long learning in an environment where an extreme amount of information permeates a discussion.

Readings

http://eprints.qut.edu.au/archive/00005398/01/5398.pdf Duffy, Peter and Bruns, Axel (2006) The Use of Blogs, Wikis and RSS in Education: A Conversation of Possibilities.

http://edtechpost.wikispaces.com/PLE+Diagrams. This amounts to a textbook of current thinking in the PLE world. It is a list of models that educational thinkers from around the world have put together to describe their own learning and how they think others might like to have their learning structured.

Activities:

Forum discussion - Choose three of the PLE diagrams (with accompanying websites) and compare the different views on Personal learning. Try to get inside the thinking of the given authors to understand why they have chosen to work in this way. What can you take away from each of the example to help construct your own PLE?

Assignments:

Start your PLE diagram, post image somewhere on the web with the course tag. (for the diagram, you can use any graphic software - Fireworks, PowerPoint, or open source tools like CMAP, or web-based tools like Gliffy)

Start your own PLE. Make some initial decisions about how you are going to create your own learning space.

Tool to explore this week: Blogs

Take a few minutes to review the blog resource page. Go to a blog service (worpress.com, blogger, or something similar) and set up a blog. If you are concerned about privacy, use a fake name/title. Experiment with posts - text, formatting, an image, etc. As you progress in the course, you can use this blog as a means of sharing your files/resources with others.

Optional: If you're interested in additional resources on PLEs, recordings from a recent online conference are available here.

Week 4: Affordances: Defining social media
According to psychologist James Gibson, an affordance is the "action potential" of a tool (or, perhaps more broadly, an environment or system). What does this mean? Essentially, it means that tools have certain uses to which they are broadly oriented. I can, for example, use a hammer to drive a nail into a board when building a house. This "action potential" of a hammer is more useful than using a hammer to clean a window. While we can argue and debate the exact action potential of different tools, we can broadly recognize appropriate and inappropriate applications.

From the perspective of this course, social media has certain affordances for learning (I should state that media itself is not inherently social, but that it becomes social as we use it for social purposes).

Defining Social Software Affordances provides a framework for viewing emerging technologies. Six key affordances can be noted in emerging technology. We will explore each of these in more detail in the coming weeks. The affordances are:

 * Access. This affordance includes many of the most popular tools in use today: Google (access information), Facebook (access people) resources
 * Declare or state presence (as currently online or in declaring physical proximity through GPS). Presence is a declaration of "what I am doing" or "where I am".
 * Expression through tools such as Second Life or profile features of most social networking site. Expression is a declaration of "this is who I am".
 * Creation of new content and resources through blogs, wikis, podcasts, Second Life
 * Interaction with others through asynchronous and synchronous tools like discussion forums, Twitter, Skype, ELGG
 * Aggregation of resources and relationships through Facebook, iGoogle, or NetVibes.

Most tools will have more than one affordance. For example, Twitter has the affordance of interaction and presence. Blogs have the affordance of creation and interaction. Google Reader - with its recent updated features of sharing with friends - has the affordances of aggregation and interaction.

Activities:

Once you've reviewed the affordances of social media link above, review:

Social Media Examples

Go to the discussion forum for week 4 and share your definition of social media. Do you agree with the discussion of media affordances listed above? What technological affordances are most critical for your school or university in Africa?

Week 5: Open and Networked Learning
As you're likely aware by this stage in the course, the tools we've considered are essentially based about connections. This is hardly surprising...after all, the internet is the backbone of the tools and concepts driving discourse in society.

How are networks expressed? At two prominent levels: how we access content and how we connect with each other.

At a content level, tools like Google, Google Scholar, Diigo, del.icio.us, YouTube, online forums, Google Groups, etc. enable relatively simple access to content. Think for a moment about your experiences as a learner as little as ten years ago. Something as simple as conducting a literature review required days (weeks?) in a library digging through stacks of journals. Now, a few hours on Google Scholar or similar academic search engine, provides a huge array of resources. Combined with open access publishing, numerous additional opportunities exist for research and content access.

In terms of interaction, the connective aspects of the internet are prominent: email, skype, instant message, twitter, and many, many others. Geographical separation is no longer the limiting element it was ten years ago.

The duality of content/interaction connections fits under the broad banner of networked learning. This week, we'll spend time exploring this concept in greater detail.

Watch the following video:

Networked Student

Activity:

 * Create a delicious account
 * View this short video on social bookmarking in plain English
 * Search for resources on "networked learning" and tag your three favorite resources with IET_F09
 * Download Jing or Articulate and play around with creating a sample presentation. Articulate is a great tool, but expensive. You can download a trial version to experiment with it...
 * Share your thoughts in the discussion forum...

Week 6: Tools for Interaction
In week 4, brief consideration was given to the concept of affordances of tools. The term affordances comes from psychology - particularly the work of James Gibson. An affordance is essentially the "action potential" of a tool - i.e. what it can be used to do/achieve/accomplish. For example, a hammer is much better suited for the action of pounding nails into wood than it is for cleaning windows.

Software also has an affordance component. When considering Skype, Moodle, Google Wave, or similar new software, educators can place the tools in different contexts by asking "what is the action potential of ____?".

This week, we will look at tools of interaction. These tools can be synchronous (real time such as skype) or asynchronous (time delayed, such as this moodle forum). Obviously, tools can have various affordances. Skype, for example, can be used for real time interaction...but it can also be used for declaring presence (i.e. the "i'm online" green dot beside a skype user's name). Or it can be used to create a podcast interview.

Activities this week:

1. Continue working on your draft presentation - due in two weeks! 2. Tag tools and resources on delicious or diigo relating to our topic "tools of interaction". Please use the course tag: AfricaIET09 3. In the moodle forum, share the two tools you have found most valuable for interaction (one synch, one asynch). 4. Create an account on slideshare (mine is here if you're interested). If you have time, upload a short powerpoint presentation, record narration with Audacity, and upload/synch both. 5. Share your thoughts on tools of interaction in the moodle forum:

Week 7: Tools for Creation
Week 6 focused on tools of interaction. This week, we'll look at a variety of tools for content creation.

One of the prominent affordances of the internet is the ease and low cost of content creation: video, audio, images, and other digital content.

Activities:

Review the following resources:

Blogs

Wikis

Podcasts

Once you've reviewed the resources, create a blog (if you haven't done so already), a wiki (wikispaces is a great resource for setting up a free wiki, but as the wiki page lists, other options are available), or a podcast.

Please post your response in the discussion forum.

Continue working on your mid-course presentation. If you have not done so yet, please post your draft presentation for others to provide comments and suggestions.

Week 8: Final Presentation Week
This is the presentation week for the course. Please post your presentation, as detailed in the course outline, to the discussion forum in Moodle...

A few examples of presentations posted by learners in previous courses:

http://www.scribd.com/doc/11531055/Flickr-and-New-Media-Literacy

http://www.screentoaster.com/watch/stVkJcQEVLRlxfQFVVW1pe/delicious_final http://rationalbob.podbean.com/2009/02/12/post-production-camstudio/

http://screenr.com/e0N

Week 9: Tools for Sharing
The online environment is about sharing - webpages, blogs, podcasts, images, etc.

The tools you'll encounter this week focus on sharing content online. Numerous tools exist, and you've used several already for sharing your podcast to sharing your thoughts on you blogs. Earlier in the course, we looked at the affordances of technology - namely, what is the "action potential" of a particular tool. Each tool has multiple affordances, resulting in some overlap (blogs can be used for creating content as well as for sharing).

Review these resources:

http://ltc.umanitoba.ca/wikis/etl/index.php/Image_sharing

http://ltc.umanitoba.ca/wikis/etl/index.php/Social_bookmarking

If you do not have an account with flickr and delicious (or diigo), please create an account. Post a few images and tag a few articles on delicious (or diigo). What are the educational uses of these tools? How could you use these in existing classrooms?

...and, if you have time, create an account with http://voicethread.com/. VoiceThread is an interesting tool that combines sharing (images, documents, and videos) with interaction through audio and text comments.

Week 10: Tools of Control: Aggregating content/conversations
At this stage in the course, you're probably feeling a bit overwhelmed with the amount of tools you've encountered and the amount of information you need to manage. Once you start reading a few blogs...and following a few journals...and a few news sites...it doesn't take long until you feel like your on the losing end of an information war.

Fortunately, there is an option to cope and make sense of this abundant information: Aggregation. Most traditional information sources, such as newspapers and journals, provide structure and the type of content they feel their readers will enjoy. With online aggregation, you are essentially "rolling your own newspaper". You decide and subscribe to the information sources you find worthwhile.

Activities this week:

1. Read this http://ltc.umanitoba.ca/wikis/etl/index.php/Aggregation

2. Create an account with one of the aggregators listed in the reading (I use Google Reader...it's a good basic starting tool).

3. Subscribe to a few blogs and news sites (details on how to do this are listed in the video in the aggregation page).

4. What do you think? Are RSS readers something you could get used to? What do you like? Don't like? Share your thoughts in the forum...

Week 11: Tools of Presence
"Tools of presence" allow us to let others know where we are...and what we are doing. While related to "tools of expression", unique elements exist. When expressing ourselves through Google or Facebook profile, we are essentially letting others know what we are about. When declaring our presence to others, we are letting them know where we are and what we are doing. Privacy issues are a big concern with presence tools. On the one hand, when we reveal our location/interests to others, we open ourselves to connections.

On the other hand, when we are open, we are also at increased risk of identity theft, harassment, or related concerns. Careful consideration is warranted when using tools of presence...

Tools to consider:

Twitter - http://www.youtube.com/watch?v=ddO9idmax0o

Set up an account, add a few friends (my id=gsiemens). Facebook - status updates...do you use these? If you are on FB and regularly update your status, have a look through the types of status updates you have posted, and ones that your friends have posted. Do you notice any privacy concerns? If so, provide a brief summary of your experience in the forum this week...

Latitude - watch this short video: http://www.youtube.com/watch?v=Q-Oq-9enE-k.

Reactions? At what point does our ability to connect move out of the realm of "cool!" and into "oh, that's a bit freaky"?

Week 12: Trends and Future Patterns
The chaotic pace of change in technology and society presents educators with a critical problem: how do we recognize important trends that have the potential to influence education? Abundance - as we've addressed at several stages in the course - obscure the consequential.

In times of change, every individual needs basic skills in tracking trends and informally (at minimum) analyzing potential impact. But how do we do this? How do we get started? That's our topic for this week.

Resources:

http://www.connectivism.ca/?p=168 Struggling for a metaphor of change - a fairly broad look at trends and a suggested model/approach to analyzing which trends are consequential

http://trendwatching.com/ - this is a useful site for societal trends. They compile a monthly theme of an important development and provide a fairly in-depth analysis of that theme.

http://www.slideshare.net/thecleversheep/ten-trends-rmc 10 Trends - slideshare presentation - on learning 2.0

After reviewing resources and readings for the week, take a look back at your experience in this course. Which trends are more significant? Which trends have had the greatest influence on you and your work? Reminder: we will be meeting in SecondLife on Tuesday. I'll send out the SLURL via email announcement. This is an optional session...

Introduction_aux_technologies_émergentes

title: Intro Emerging Tech


text:
User talk:B9 hummingbird hovering/༄༅༅༎ རྫོགས་པ་ཆེན་པོ་ཀོང་ཆེན་སྙིང་ཐིག་གི་སྔོན་འགྲོའི་ངག་འདོན་ཁྲིགས་སུ་།

100 syllable mantra of Vajrasattva ཨོཾ་བཛྲ་སཏྭ་ས་མ་ཡ་མ་ནུ་པ་ལ་ཡ། བཛྲ་སཏྭ་ཏྭེ་ནོ་པ་ཏིཥྛ། དྲྀ་ཌྷོ་མེ་བྷ་བ། སུ་ཏོ་ ཥྱོ་མེ་བྷ་བ། སུ་པོ་ ཥྱོ་མེ་བྷ་བ། ཨ་ནུ་ར་ཀྟོ་མེ་བྷ་བ། ས་རྦ་སི་དྡྷི་མེ་པྲ་ཡ་ཙྪ། ས་རྦ་ཀ་རྨ་སུ་ཙ་མེ ཙི་ཏྟཾ༌ཤེ་ཡཿ་ཀུ་རུ་ཧཱུྂ། ཧ་ཧ་ཧ་ཧ་ཧོཿ བྷ་ག་བ་ན ས་རྦ ཏ་ཐཱ་ག་ཏ་བཛྲ་མཱ་མེ་མུ་ཉྩ། བཛྲཱི་བྷ་བ་མ་ཧཱ་ས་མ་ཡ་སཏྭ ཨཱཿ །། ཧཱུྂ ཕ་ཊ

Seed syllable
༼དྷཱིཿ༽  dhI.h in Tibetan Uchen.


text:
User:Benyapha

Assignment 4: Operating System  Question 1) Look at different operating system (how many use it?)   Answer :  Different types of operating systems:  1. Microsoft Windows  plays a very big role in the market of personal computers by 91%. Also, there are nine main versions of Microsoft Windows, which are Windows’95, Windows’98, Windows’ 2000, Windows NT, Windows ME, Windows XP, Windows CE, Windows VISTA, and Windows 7. 2. Linux  is usually used to install in embedded systems, mobile phones, some kinds of watches, desktop computers, laptop computers, netbooks, and supercomputers. Nowadays, Linux has been used to be an operating system in netbooks with the two kinds of devices, which are ASUS Eee PC and Acer Aspire One. Brazil, Russia, China, India, Spain, France, and Germany has been using Linux for various kinds of purposes. For example, Linux has been developed to be used in educational organizations and other official organizations. 3. Mac OS X has been developed and owned by Apple Inc. Mac OS X became the second most famous operating system in 2009 in terms of the usages of the Internet connection. Also, Mac OS X includes many different languages. For example, English, Japanese, French, German, Spanish, Portuguese, and Italian are available. There are four main types of operating systems, which are “Real-time Operating Systems”, “Single-User/Single-Tasking Operating Systems”, “Single-User/Multitasking Operating Systems”, and “Multi-User/Multitasking Operating Systems”. •	The first type of operating system is “Real-time Operating Systems” or “RTOS”. RTOS is also known as Embedded Operating Systems. A real-time operating system runs real-time application. Also, a real-time operating system support only single-tasking. A real-time application is an application that responses to the input extremely fast. BeOS, ChorusOS, eCos, FreeRTOS, ITRON, LynxOS, MicroC/OS-II, Nucleus, Nut/OS, OS-9, OSE, OSEK/VDX, OSEKtime, pSOS, QNX, RMX, RSX-11, RT-11, RTOS-UH, VRTX, VxWorks, Windows CE, RTLinux, and RTAI are examples of Real-time Operating Systems. Real-time applications are used in medical diagnostics equipment, life-support systems, machinery, scientific instruments, and industrial systems. •	The second type of operating system is “Single-User/Single-Tasking Operating Systems”. This type of operating system allows users to run a single task at a time. Printing a document, writing a file to disk, editing a file, and downloading a file from a network server are examples of task for a computer user. A small and simple operating system can manage only a time. MS-DOS is an example of single-user/single tasking operating systems. This type of operating system is usually found in the Palm handheld computers. •	The third type of operating system is “Single-User/Multitasking Operating Systems”. This type of operating system allows users to run many tasks at a time. Microsoft Windows and the Macintosh Operating System are the examples of this type of operating system. This type of operating system is usually found in personal computers. Also, it can be used in various kinds of work. •	The forth type of operating system is “Multi-User/Multitasking Operating Systems”. This type of operating system allows users to run on a terminal server. Each user’s applications run within their user session on the server separate from all other user sessions. A terminal client is the software that makes a user session possible. UNIX, VMS, and MVS are examples of multi-user/multi-tasking operating systems. MVS is used in mainframe operating systems.

 Question 2) What operating systems should install on your netbook?   Answer : Windows 7 and Linux are the two popular operating systems that are used in netbooks. Nowadays, Windows 7 is recommended to install on many new versions of netbooks because of the three following reasons. The first reason is that Windows 7 can provide lifetime for netbooks longer than previous operating systems. The second reason is that netbook users will be able to access into the network more convenient. The third reason is that Windows 7 does not need power to run machines compared to other previous versions of operating systems. Acer Aspire One D250, EasyPC E790, Lenovo Ideapad S12, Toshiba Satellite U505 and M505, HP Mini 311, gScreen, iPhone 3G, Sony VAIO X, and Samsung N130, N140, and N150 are examples of netbooks which use Windows 7 as an operating system. 2.1) Acer Aspire One D250 is a netbook that provides the two operating systems, which are “Windows” and “Android”. 2.2) EasyPC E790 is a netbook that provides only one operating system, which is “Windows CE 5.0” or “Androud”. 2.3) Lenovo Ideapad S12 is a netbook that provides only one operating system, which is Windows 7. 2.4) Toshiba Satellite U505 and M505 are the two versions of netbook that provide Windows 7 as an operating system. 2.5) HP Mini 311 is a netbook that provides only one operating system, which is Windows 7. 2.6) gScreen is a netbook that provides only one operating system, which is Windows 7 Home Premium. 2.7) iPhone 3GS is a netbook that provides only one operating system, which is iPhone OS 3.1.2 (build 7D11), released October 8, 2009. 2.8) Sony VAIO X is a netbook that provides only one operating system, which is Windows 7. 2.9) Samsung N130, N140, and N510 are netbooks that provide Windows 7 Starter Edition.

Assignment 3: Twitter http://twitter.com/Benyaphaben '''Question 1. How is Facebook and Twitter the same?''' Answer: The first similarity between Facebook and Twitter is that a user can update a current status. The second similarity between Facebook and Twitter is that a user can find friends and add/follow friends via the tool of Facebook and Twitter. The third similarity between Facebook and Twitter is that a user can post a picture into those websites. Question2: Why do you think Facebook is worried about competition from Twitter? Answer: Perhaps the registration process of Twitter is not that complicated like Facebook. On Twitter, a user fill the personal information in a few boxes, but on Facebook, a user has to fill the personal information in many boxes. Also, the functions of the Twitter website can be used very easily. Therefore, these two reasons may be an answer for this question. A twitter user can follow many main news reports from BBC, NYTIMES, and Financialtimes very easily. A twitter user will be able to hfollow celebrities and Prime Misters of Thailand or the United States of America. Subscript text

Assignment 2: Best of the BestApple iMac "Core 2 Duo" 3.06 24-Inch (Early 2009) Specs (MB420LL/A)

1.Introduction Date: March 3, 2009

2.Discontinued Date: N/A

3.Processor: All Intel “Core 2 Duo” processors include two independent processor “cores” on a single silicon chip. 3.1 Processor Type: Core 2 Duo (E8435) 3.2 Processor Speed: 3.06 GHz 3.3 Processor Upgrade: ZIF Socket 3.4 FPU: Integrated

4. System Bus Speed: 1066 MHz

5. Cache Bus Speed: 3.06 GHz (Built-in)

6. ROM/Firmware: 6.1 ROM/Firmware Type: EFI 6.2 ROM/Firmware Size: N/A

7. L1 Cache and L2/L3 Cache: 7.1 L1 Cache: 32k/32k 7.2 L2/L3 Cache: 6 MB (on chip)

8. RAM: Supports 1066 MHz PC3-8500 DDR3 SO-DIMMs. 8.1 RAM Type: PC3-8500 DDR3 8.2 Min. RAM Speed: 1066 MHz 8.3 Standard RAM: 4 GB 8.4 Maximum RAM: 8 GB 8.5 Motherboard RAM: None 8.6 RAM Slots: 2

9. Video Card: GeForce GT 130 -This model has a NVIDIA GeForce GT 130 with dedicated GDDR3 memory for video.

10. VRAM 10.1 VRAM Type: GDDR3 10.2 Standard VRAM: 512 MB 10.3 Maximum VRAM: 512 MB

11. Display: 11.1	Built-in Display: 24.0” Widescreen 11.2	Native Resolution: 1920x1200 11.3	2nd Display Support: Dual/Mirroring 11.4	'''2nd Max. Resolution''': 2560x1600

12.	Standard Hard Drive: 1 TB (7200 RPM)

13.	 '''Int. HD Interface''': Serial ATA

14.	 Standard Optical: 8X DL “SuperDrive”

15.	Standard Disk: None

16.	 Standard Modem: None

17.	Standard Ethernet: 10/100/1000Base-T

18.	 Standard AirPort:  802.11a/b/g/n

19.	 Standard Bluetooth: 2.1+EDR

20.	 USB Ports: 4 (2.0) 21.	Firewire Ports: 1 (800) 22.	 Expansion Slots: None 23.	 Expansion Bays: None 24.	 '''Incl. Keyboard''': Apple Keyboard* 25.	 '''Incl. Input''': Mighty Mouse 26.	 Case Type: All-in-one 27.	 Form Factor: iMac Core 2 Duo (2009) 28.	 Apple Model No: MB420LL/A 29.	  Apple Subfamily: Early 2009 30.	 Gestalt/Model ID: N/A 31.	 Machine Model: iMac9,1 32.	 Battery Type: N/A 33.	 Battery Life: N/A 34.	  Pre-Installed MacOS: X 10.5.6 (9G2030) 35.	 Maximum MacOS: Current 36.	 MacOS 9 Support: None 37.	 Windows Support: Boot/Virtualization 38.	 Dimensions: 20.5 x 22.4 x 8.1 39.	 '''Avg. Weight''': 25.4 lbs. (11.5 kg) 40.	Original Price: US $2199 41.	 Est. Current Retail: US $2199

Assignment 1: Create a wiki university account → This is the first Computer Essentials homework. → The instructor, Mr. Brian Phillips, assigned this work since Monday 21st September, 2009. → This work is due on Monday 28th September, 2009.

title: User:Benyapha


text:
User:Arieli Hanner

Arieli Hanner
AP English Language

Units
Education

title: User:Arieli Hanner


text:
User:Brianna Iowa

Brianna Iowa
AP English Language

Units
Education

Nonfiction - Books

 * Angelou, Maya. A Song Flung Up to Heaven . New York: Random House, Inc., 2002.

Summary: A Song Flung Up to Heaven is an autobiography of Maya Angelou's struggle to success. Her story covers her childhood, teenage years, and her as an adult. Within her story one will learn more about her family and friends and the life she lived. Her times of brutality only made her stronger so that now she will not allow any obstacles to interfer in her future goals.

"We delighted in being big city kids until my mother's boyfriend raped me". "I wrote the first line in the book, which would become I Know Why the Caged Bird Sings ". --Brianna Iowa 13:51, 1 October 2009 (UTC)

title: User:Brianna Iowa


text:
User:Odelia Kadi

Ms. Odelia Kadi
AP English Language

Units
Education

Readings, Observations, and Experiences
Whether for class or my own enjoyment, these are a collection of some of my sources for learning.

Nonfiction - Books

 * Walls, Jeannette. The Glass Castle . New York: Scribner, 2005.

Jeannette and her family are always struggling to find enough money to live off of. They have a dream that one day they will find gold and be able to live the lifestyle of a millionare. Even though they know that the chances that they will find gold are nearly impossible, they never loose faith and they alway remain with each other. However, her fathers alcohol problem doesn't make their economic struggle any easier. "We fought over who got to sleep with the dogs-Tinkle, the Jack Russell terrier, and Pippin, a curly-haired mutt who had wandered down through the woods one day-becaue they kept us warm."

"We couldn't afford wood any more than we could afford coal, and Dad wasn't around to chop and split any, which meant it was up to us kids to gather dead branches and logs from the forest." --Odelia Kadi 13:06, 2 October 2009 (UTC)Odelia Kadi

title: User:Odelia Kadi


text:
User:Luisa Nomad

Luisa Nomad
AP English Language

Units
Education

Nonfiction - Books

 * Walls, Jeannette. The glass castle. New York; Jeannette Walls, 2005.

The glass castle is a memoir of Jeannette walls. The memoir discuss  her childhood life in poverty and Walls had to defend for herself and siblings due to the neglecting of her intoxicated father and mother that rather paint and write than take care of her responsibilities of  her house hold. Eventually Walls and her siblings found the courage to leave their parents and be on their own. Jeannette Walls is now a successful writer and is happily married to John Taylor.

"we moved from city to city trying to find a place to stay". --Luisa Nomad 13:20, 2 October 2009 (UTC)

title: User:Luisa Nomad


text:
User:Sperry Nicholas

Mr. Sperry Nicholas
AP English Language

Units
Education

title: User:Sperry Nicholas


text:
Introduction to psychology/Psy102/Tutorials/Cognitive processes and intelligence/Full problems

This page contains the full problems and answers for the tutorial about cognitive processes and intelligence. Do not read this page if you want to participate in the class tutorial first.

Bat and ball: Limits of intuition
Frederick (cited by Kahneman, 2003) suggests a simple puzzle to demonstrate the limits of everyday intuition:
 * 1) Question: “A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?”
 * 2) Most people report an initial tendency to answer “10 cents” because the sum $1.10 separates easily into $1 and 10 cents and because 10 cents is about the right magnitude.
 * 3) Frederick reported that 50 percent of Princeton students and 56 percent of University of Michigan students gave this wrong answer.
 * 4) Simple subtraction convinces students of their error: $1.00 for the bat – $0.10 for the ball = $0.90, not $1.00. The correct answer is $1.05 for the bat, $0.05 for the ball.
 * 5) What's going on? Why?

Train stations: Mental set
This exercise illustrates the limitations of having a mental set. A mental set refers to an approach to problem solving which you have learnt from past experience. If we are presented with a different kind of problem, our mental set is likely to be used and we have difficulty finding an answer.

Ask students to get out a pen and paper which they will need in order to solve a problem. Then read the following narrative slowly, pausing between numbers and sentences:

"# Assume that you’re the engineer of a passenger train.
 * 1) At the first station, 20 passengers get on.
 * 2) At the next station, 5 passengers get off and 15 get on.
 * 3) At the next station, 10 passengers get off and 12 get on.
 * 4) At the next station, 7 get off and 10 get on.
 * 5) At the next station, 20 passengers get off and 5 get on.
 * 6) At the next stations, 8 passengers get off and 3 get on."

After reading the story, ask students to write down answers to the following questions:
 * 1) How old is the engineer of the train?
 * 2) How many stations were there?
 * 3) How many passengers are left on the train?
 * 4) Altogether, how many passengers have disembarked the train since the first station?
 * 5) Altogether, how many passengers have boarded the train along its route.

Answers
 * 1) 1st question might startle a lot of students who might claim it is unanswerable, but just repeat the first sentence and they will groan.
 * 2) 2nd question will be difficult unless they have heard the problem before. We are so busy attending to changes in the number of passengers that we don’t attend to the stop themselves (mental set).
 * 3) 3rd question - most will get correct as their mental set had them continually processing passenger total.
 * 4) 4th and 5th questions - will probably be missed because the students will have kept to one “set” ie keeping track of the revelant quantities.

Conclusion
 * 1) It is almost impossible to process all the information needed to answer the questions because we have a limited capacity to store and process new material. Keeping track of more than one or two of the relevant figures creates information overload, so we tend to use specific strategies that have worked in the past.
 * 2) Consider that this mental sets and our limited information processing capacity relates to many other aspects of psychology, e.g., eyewitness testimony and intelligence testing.

Linda: Representativeness heuristic
Read the following to students:

"Linda is 31, single, outspoken, and very bright. She majored in philosophy at university. As a student, she was deeply concerned with discrimination and other social issues, and she participated in antinuclear demonstrations."

Then ask them to indicate whether they think statement 1 or statement 2 is more likely.
 * 1) Linda is a bank teller.
 * 2) Linda is a bank teller and active in the feminist movement.

Did the majority choose statement 2? Ask them why and get them to discuss it.

We tend to judge the likelihood of things in terms of how well they seem to represent or match a particular prototype. While “representativeness” works much of the time, it leads to errors when its conclusion runs counter to the laws of chance. You can explain that the probability of any two uncertain events occurring together (i.e., being a bank teller and being active in the feminist movement) is always less than the odds of either happening alone.

Buddhist monk problem
Read the following problem to students and encourage them to discuss how they might solve it.

"One morning at sunrise a Buddhist monk began to climb a mountain on a narrow path that wound around it. He climbed at a steady 3 miles per hour. After 12 hours he reached the top where there was a temple and remained there to meditate for several days. Then, at sunrise he started down the same path, walking at a steady 5 miles an hour. Prove that there must be a spot along the same path which he occupied on both trips at exactly the same time of day."

- 30px

The Buddhist monk problem demonstrates how some problems are most easily solved by representing them graphically or visually rather than trying to work it out mathematically.

Graphically, try drawing this as one monk climbing and one monk descending (from side-on - altitude is the Y axis and time is the X axis). At same point, the lines (paths) must cross. Visually we can't tell at exactly what time, but the question simply asks to prove that there must be a spot he occupied at the same time on both trips. Regardless of the two monks' walking rates, the two paths would have to have cross at some point. Now simply pretend these two trips happened on two separate days – the monk was at a same spot on the path at precisely the same time of day for both trips. To view a graph, see Slide 8 of this powerpoint file

Source: Figure from Cognition (4th ed.), by Margaret Matlin, p. 352.

Overconfidence phenomenon

 * 1) Distribute the truth/lie handout to students and explain you are going to test their ability to distinguish truth from lies.
 * 2) Explain that you have put 10 slips of paper in an envelope, 5 of which say “tell the truth” and 5 of which say “tell a lie”.
 * 3) Get 10 volunteers to take a slip each and not to let other students know which they have.
 * 4) For each topic, select a volunteer from one of the 10 with the slips of paper, and get that student to stand up and tell his/her truth or lie. The remainder of the students write down whether they thought it was the truth or a lie and how confident they are (0 – 100%). When all the stories are told, the volunteers reveal which statements were truthful and which were lies.
 * 5) Then get students to compute the number they got correct out of 10, and their average confidence level (you might have to explain how to do this).
 * 6) Then ask how many were more correct than confident (should only be a few) and then how many were more confident than correct (should be the majority). If students just guessed the answers, they should be right 50% of the time because half were telling the truth and half were lying.
 * 7) This demonstrates our tendency to overestimate the accuracy of our current knowledge.
 * 8) Milojkovic and Ross reported their Stanford students were 52% correct and 73% confident – similar to the levels achieved using these materials. Milojkovic and Ross also report that when people were 90 to 100% confident, they were no more correct than when they were only 50 to 65% confident.
 * 9) See also overconfidence effect (Wikipedia)

What is intelligence?

 * 1) Invite the class to define intelligence (this should be difficult - try to elicit different types of definitions, looking for common themes). Some ideas which may come up:
 * 2) Knowledge? (Debatable - although this was commonly tested esp. in earlier IQ tests)
 * 3) What IQ tests measures (A somewhat cynical, circular definition)
 * 4) Mental speed, accuracy, efficiency, and underlying capacity to solve problems
 * 5) One intelligence - or many?
 * 6) Intelligence is a subset of cognition - i.e., those cognitive processes which combine to form  mental ability to solve different kinds of problems
 * 7) Intelligence is the single most tested/measured psychological quality (with personality not far behind) - study and testing of intelligence it has also played an important role in psychology becoming a science
 * 8) But intelligence testing is fraught with potential dangers, problems, and controversies - more about this in the lecture next week...

Intelligence line-ups
Clear desks and tables to the side, then ask the class to line-up in order of how much they believe each of the following (ask some people for each line-up where they sit and why):
 * 1) Intelligence is due to nature vs. nurture
 * 2) Intelligence is single vs. plural (single, general underlying ability (g) vs. multiple, distinct abilities (multiple intelligences))
 * 3) Intelligence is speed of processing vs. knowledge you have
 * 4) Intelligence is related to whether one was breast-fed or not
 * 5) Intelligence is increased by school attendance school vs. not related to school attendance
 * 6) IQ is stable vs. rising
 * 7) Intelligence predicts or doesn't predict income.
 * 8) Intelligence can vs. cannot be artificially created (artificial intelligence).
 * 9) IQ is affected by diet vs. is not affected

Many of the intelligence line-ups are discussed in 12 interesting facts about intelligence described by Ceci (2001) :

 * 1) IQ is associated with some simple abilities. No one with measurable IQ has difficulty deciding which of two lines is longer or whether two pairs of letters are identical. However, in order to perform these simple tasks, a person with an IQ below 70 may need up to five times longer than an individual with a higher IQ. The nervous systems of those with low IQs are simply less efficient.
 * 2) School attendance correlates with IQ. Staying in school can elevate IQ or, more accurately, keep it from slipping. Evidence for this dates back to the turn of the twentieth century when the London Board of Education found that the IQs of children in the same family decreased from the youngest to the oldest. The older children progressively missed more school. Toward the end of the Vietnam War, a lottery determined draft priority. Those men born on July 9, 1951, were picked first so they tended to stay in school longer in order to avoid the draft. Those men born July 7 were last in the lottery and thus had no incentive to stay in school. Men born on July 9 had higher IQs and also earned 7 percent more money. Summer vacations also seem to affect IQ. With each passing month, children’s end-of-year scores decline.
 * 3) IQ is not influenced by birth order. The idea that birth order influences personality and intelligence has not stood up under recent scrutiny. Moreover, the claim that large families make low-IQ children may be unfounded because researchers have found that low-IQ parents make large families. Smart people tend to have small families, but it is not small families per se that make people smart.
 * 4) IQ is related to breast-feeding. Even when researchers control for factors such as the sense of closeness mother and child experience through nursing, breast-fed children appear to have an IQ of 3 to 8 points higher by age 3.
 * 5) IQ varies by birth date. State policies mandate the age of students entering school as well the age they may leave, typically 16 or 17. Those born in the final three months of the year are more likely to enter school a year later; thus, when they leave school, they have been attending one year less. For each year of school completed, there is an IQ gain of approximately 3.5 points. Unsurprisingly, as a group, those born later in the year show a lower IQ score.
 * 6) IQ evens out with age. Imagine, suggests Ceci, two biological siblings adopted by two different middle class families, at age 5 and again in early adulthood. Are their IQs more alike when younger and living in the homes of their adoptive parents or when they are older and living on their own? Contrary to expectation, as the siblings go out on their own, their IQ scores become more similar. The probable reason is that once they are away from the dictates of their adoptive parents, they are free to let their genotypes express themselves. Because they share about 50 percent of their genes, they will become more alike because they are likely to seek similar sorts of environments.
 * 7) Intelligence is plural, not singular. Regardless of their views of so-called general intelligence, researchers agree that there are statistically independent mental abilities such as spatial, verbal, analytical, and practical intelligence. Howard Gardner is, of course, a primary proponent of multiple intelligences theory.
 * 8) IQ is correlated with head size. Modern neuroimaging techniques demonstrate that cranial volume is correlated with IQ. Evidence also comes from studies of the helmet sizes of members of the Armed Forces whose IQs were measured during basic training. Correlations are quite small.
 * 9) Intelligence scores are predictive of real-world outcomes. Even among those with comparable levels of schooling, the greater a person’s intellectual ability, the higher the person’s weekly earnings. Those with the lowest levels of intellectual ability earn only two-thirds the amount workers at the highest level earn.
 * 10) Intelligence depends on context. In visiting racetracks, researchers found that some men were excellent handicappers while others were not. A complex mental algorithm that was used to convert racing data from the racing programs sold at the track distinguished experts from nonexperts. However, the use of the algorithm was unrelated to the men’s IQ scores. Some experts were dockworkers with IQs in the 80s, but they reasoned far more complexly at the track than all nonexperts, even those with IQs above 120. At the same time, the experts performed very poorly at reasoning outside the track.
 * 11) IQ is going up. IQ has risen about 20 points with every generation, an increase called the "Flynn effect," after New Zealand political scientist James Flynn. The rise in IQ has been attributed to better nutrition, more schooling, and better-educated parents.
 * 12) IQ may be influenced by the school cafeteria menu. In one large study, 1 million students enrolled in the New York City school system were examined before and after preservatives, dyes, colorings, and artificial flavors were removed from lunch offerings. The investigators found a 14 percent improvement in IQ after the removal. Improvement was greatest for the weakest students.

World War I IQ Test

 * 1) WWI Army Intelligence Test - Answers
 * 2) Note that an intelligence test does not measure drive, persistence, creativity or any of the myriad other skills that often count for more in achieving success out of school.
 * 3) A low score on an IQ test does not mean probable failure in life. All it means is that the person taking the test did poorly on that particular test. If the score is a valid reflection of cognitive ability, then the person will probably find particularly complex cognitive tasks harder than "normal".
 * 4) However, most of us do not spend our lives in situations that can be easily measured by paper-and-pencil tests. Thus, scores obtained on such tests should be viewed with some restraint and skepticism, particular if they are high or low.
 * 5) IQ tests measure only one aspect of overall "life skill".

Chitling test of intelligence

 * 1) Handout out the The Chitling Test of Intelligence - ask students to complete and then score according to this answers.
 * 2) Formally the Dove Counterbalance General Intelligence Test, the Chitling Test of Intelligence was designed by Adrian Dove, an African-American sociologist, as a half-serious attempt to show that IQ tests are culturally biased in part due to American children are just not all speaking the same language.