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Math 34 Differential Equations Exam \#1
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                               October 9, 2003 \hfill  SHOW ALL
WORK
~~~

Solve the following differential equations

[15]~  1a.) $t^2y' + 2ty = t~sin(t)$.

\centerline{Answer 1a.) $\underline{\hskip 3in}$}


[15]~  1b.) $y'' - 4y' + 4y = 0$, $y(0) = 2$, $y'(0) = 3$.

\u
\centerline{Answer 1b.) $\underline{\hskip 3in}$}
\w

[15]~  1c.) $y'' - 4y' + 4y = 5t + 1$.
\u
\centerline{Answer 1c.) $\underline{\hskip 3in}$}

[15]~  1d.) $2y'' - 3y^2 = 0$, $y(0) = 1$, $y'(0) = 1$..


\centerline{Answer 1d.) $\underline{\hskip 3in}$}

[5]~ 2.)  Use Euler's formula to write $e^{2 + 3i}$ is the form of $a + ib$.
\vskip 1.7in
\centerline{Answer 2.) $\underline{\hskip 3in}$}


[5]~ 3.)  Draw the direction field for $y' = y$.
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4.)  Circle T for true and F for false.

[3]~ 4a.)  Suppose 
$\psi_1$
 and
$\psi_2$
are solutions to the linear equation, 
$ay'' + by' + cy = g(t)$, 
then 
$\psi_1 + \psi_2$ must also be a solution to
$ay'' + by' + cy = g(t)$. 

\rightline{T~~~~~~~~~~~F}
\w

[3]~ 4b.)  Suppose 
$\psi_1$
 and
$\psi_2$
are solutions to the equation, 
$ay'' + by' + cy^2 = 0$, 
then 
$\psi_1 + \psi_2$ must also be a solution to
$ay'' + by' + cy^2 = 0$. 

\rightline{T~~~~~~~~~~~F}

\w

[3]~ 4c.)  Suppose 
$\psi_1$
is a solution to the linear equation, 
$ay'' + by' + cy = g(t)$, 
 and
$\psi_2$
is a solution to the linear equation, 
$ay'' + by' + cy = f(t)$, 
then 
$5\psi_1 + 3\psi_2$ must also be a solution to
$ay'' + by' + cy = 5g(t) + 3f(t)$. 

\rightline{T~~~~~~~~~~~F}
\w

[3]~ 4d.) If $p$, $q$, and $g$ are continuous, then there exists a unique solution to 

\centerline{$y'' + p(t)y' + q(t)y = g(t)$,
$y(t_0) = y_0$,
$y(t_1) = y_1$.}

\rightline{T~~~~~~~~~~~F}
\w

[3]~ 4e.) If $p$, $q$, and $g$ are continuous, then there exists a unique solution to 

\centerline{$y'' + p(t)y' + q(t)y = g(t)$,
$y(t_0) = y_0$,
$y'(t_0) = y_0'$.}

\rightline{T~~~~~~~~~~~F}
\w

[3]~ 4f.)  Given an initial value, there always exists a unique solution to any first order
differential equation.

\rightline{T~~~~~~~~~~~F}

5.)  Choose one of the two following problems.  Clearly indicate which problem you have chosen.

5A.)  Suppose the equation ${dp \over dt} = \gamma p$ describes the population of field mice.  
If the population of field mice doubles in 10 years, how long will it take the population to 
quadruple.


5B.)  Find the escape velocity for a body projected upward with an initial velocity $v_0$ from 
a point 3R above the surface of the earth, where $R$ is the radius of the earth.  Neglect air 
resistance.  Recall that the equation of motion is $m{dv \over dt} = -{mgR^2 \over (R + x)^2}$
where $x$ is the distance from the earth's surface.




\vfill
\centerline{Answer 5A or 5B.) $\underline{\hskip 3in}$}
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