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Math 34 Differential Equations Final Exam
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                               December 15, 2003 \hfill  SHOW ALL
WORK
~

Choose 10 of the following 11 problems. If you choose fewer than 10, the points for 
the 
remaining problems will be appropriately averaged.  3 of the problems have 1 point extra 
credit available (indicated by [1-EC]).  You can do the extra credit even if you do not 
choose these problems.


[3]~ 1a.)  Without using the LaPlace transform, solve the following initial value problem:  

\centerline{$y'' + 2y' + y = 0, ~y(0) = 0, ~y'(0) = 4$}

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\centerline{Answer 1a.) $\underline{\hskip 5in}$}
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[4]~ 1b.)  Without using the LaPlace transform, find the general solution to the differential 
equation
$y'' +  2y' + y = e^{-t}$

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\centerline{Answer 1b.) $\underline{\hskip 5in}$}
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[3]~ 1c.)  Without using the LaPlace transform, solve the following initial value problem:

\centerline{$y'' +  2y' + y = e^{-t}, ~y(0) = 0, ~y'(0) = 4$}

\vskip 2in
\centerline{Answer 1c.) $\underline{\hskip 5in}$}
\w

[10]~ 2.)  Use the LaPlace transform to solve the following initial value problem:

\centerline{$y'' +  2y' + y = e^{-t}, ~y(0) = 0, ~y'(0) = 4$}

\vskip .5in
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\centerline{Answer 2.) $\underline{\hskip 5in}$}
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[1-EC]~~ 2a.)  How can you use the LaPlace transform to find the general solution to a 
differential equation.

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[10]~ 3.)  Find the inverse LaPlace transform of $e^{-3s}{s \over s^2 + 8s + 18}$

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\centerline{Answer 3.) $\underline{\hskip 5in}$}
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[10]~ 4.)  Use the definition and not the table to find the LaPlace transform of $f(t) = 
3t^2$.  CLEARLY indicate when you are taking a limit.

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\centerline{Answer 4.) $\underline{\hskip 5in}$}
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[5]~ 5a.)  Find the inverse LaPlace transform ${1 \over (s-3)(s^2 - 6s + 
10)}$ by using the 
convolution 
integral.  
Leave your answer in terms of a 
convolution integral:

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\centerline{Answer 5a.) $\underline{\hskip 5in}$}
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[5]~ 5b.)  Evaluate the convolution integral obtained in 3a.

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\centerline{Answer 5b.) $\underline{\hskip 5in}$}
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[10]~ 6.) Solve the following initial value problem:  ${y' \over t^2} = -3y + 1, ~y(0) = 0$


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\centerline{Answer 6.) $\underline{\hskip 5in}$}


[10]~ 7.)  Draw the direction field for the differential equation $y' = (y-2)(y + 2)$.  
Based on
the direction field, determine the behavior of $y$ as $t \rightarrow \infty$.  If this behavior 
depends on the initial value of $y$ at $t=0$, describe this dependency.

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[5]~ 8a.)   Transform $x'' - 2x' - 3x = e^t$ into a system of first order
differential equations.

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\centerline{Answer 8a.) $\underline{\hskip 5in}$}
\w

[5]~ 8b.)  Use Euler's formula to write $e^{4 - 2i}$ in the form of $a + ib$.
\vskip 1.7in
\centerline{Answer 8b.) $\underline{\hskip 3in}$}



9.)  Circle T for true and F for false.

[2]~ 9a.)  Given an initial value, there always exists a unique solution to any second order
differential equation.

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

[2]~ 9b.)  Given an initial value, there always exists at least one solution to any second
order
differential equation.

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

[2]~ 9c.) If there is no damping and no external force, then the general solution to a 
mechanical 
vibration problem with constant spring force must be of the form $c_1cos(\omega_0t) + c_2 
sin(\omega_0t)$ 

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


[1]~ 9d.)  $y = c_1e^{3t} + c_2te^{3t} + 4cos t$ is a possible general solution to a 
mechanical
vibration problem with damping.

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


[4 or 1-EC]~ 9e.)  Explain your answer to problem 9d.
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[10]~  10a.)  A mass weighing 2 kg stretches a spring .1m.  If the mass is pulled down an 
additional 
.2m and released, and if there is no damping, determine the position of the mass at any time 
$t$

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\centerline{Answer 10a.) $\underline{\hskip 5in}$}

[1-EC]~  10b.)  Do the initial conditions affect the long-term behavior of the motion of the 
mass?

\eject
[10]~ 11.)  Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded 
continuously.  Find the time $T$ required for the original sum to double in value as a 
function of $r$. Assume that the rate of change of the value of the investment is equal to 
the interest rate $r$ times the current value of the investment $S(t)$.

\vskip 6.99in

\centerline{Answer 11.) $\underline{\hskip 5in}$}

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