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Math 34 Differential Equations Quiz 3
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WORK
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1.)  Define:  The LaPlace transform of $f = {\cal L}(f) = \underline{~~\int_0^\infty e^{-st}f(t)dt~~}$
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2a.)  ${\cal L}(0) = \underline{~~0~~}$
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2b.) ${\cal L}^{-1}({3 \over (s+1)^2 + 2}) = \underline{~~{3e^{-t} \over \sqrt{2}}sin(\sqrt{2}t)~~}$



${\cal L}^{-1}({3 \over (s+1)^2 + 2}) =$
$3{\cal L}^{-1}({1 \over (s+1)^2 + 2}) =$
${3 \over \sqrt{2}}{\cal L}^{-1}({\sqrt{2} \over (s+1)^2 + 2}) =$
${3e^{-t} \over \sqrt{2}}{\cal L}^{-1}({\sqrt{2} \over (s)^2 + 2}) =$
${3e^{-t} \over \sqrt{2}}sin(\sqrt{2}t)$

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3.)  Circle T for True or F for False:
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3a.) Suppose $y = f(t)$ is a solution to $y'' 
+ y' + y = cos(2t)$, $y(0) = 0$, $y'(0) = 0$, \break and suppose 
$y = g(t)$ is a solution to $y'' 
+ y' + y = cos(2t)$, $y(0) = 100$, $y'(0) = -200$. \break
For large values of $t$, $f(t) - g(t)$ is very small.
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{T ~~~~~~~~~}

3b.)   The initial conditions have a transient effect on the solution to  $y''  + y = cos(2t)$.

\rightline{~~~~~~~~~~F}
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4.)  Match the following differential equation initial value problem to its graph:

B = I.) $y'' + y = cos(t)$, $y(0) = 0$, $y'(0) = 0$

A = II.) $y'' + y = cos(1.2t)$, $y(0) = 0$, $y'(0) = 0$

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II = A.) \includegraphics[width=20ex]{spring2aaaplot2d2}
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I = B.)  \includegraphics[width=20ex]{spring2aaaplot2d1}
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