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Math 34 Differential Equations Exam \#2
\vskip -10pt
                               April 8, 2005 \hfill  SHOW ALL
WORK
~


[16]~~ 1.)  Find the LaPlace transform of $g(t) = 
\cases{ e^t & $t < 1$ \cr 0 & $1 \leq t < 3$ \cr 4 & $t 
\geq 3$ }$
\vskip 3.3in
\vfill

\centerline{Answer :$\underline{\hskip 5in}$}  
\eject

[16]~~ 2.)  Find the inverse LaPlace transform of ${8 \over s^2 - 3s - 4}$

\vfill
\vskip 3.4in
\centerline{Answer :$\underline{\hskip 5in}$}  

\eject



[16]~~ 3.)   Use the LaPlace transform to solve the given initial value problem.

\centerline{ $y'''  = 2, ~~~~ y(0) = 0,~~ y'(0) 
= 9,~~ y''(0) = 0$}
\vfill



\centerline{Answer $\underline{\hskip 5in}$}

\eject

[16]~~ 4.)  Use the definition and not the table to find the LaPlace transform of $f(t) = 4$.  
Show all steps including where you need to take limits.


\vfill
\centerline{Answer :$\underline{\hskip 5in}$}  



[16]~~  5.)  A mass of 30 kg stretches a spring 2m.  The mass is acted on by an external 
force of 
$8cos(2t)$
The mass is pushed upward 3m above its equilibrium position, and then set in motion in the 
downward direction 
with a velocity of 4 m/sec.  
If the mass moves in a medium that imparts a viscous force of 10 N when the speed of the mass 
is 2 m/sec, 
formulate the initial value problem describing the motion of the mass.
\vskip 1.3in
\centerline{Answer 5.) $\underline{\hskip 5in}$}




\eject

6.)   Circle T for   True and F for False.  

[4]~~	6a.)  Suppose ${\cal L}(f)$ and  ${\cal L}(g)$ exist.   ${\cal L}(fg) = {\cal L}(f) 
{\cal L}(g)  $

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

[4]~~	6b.)  Suppose ${\cal L}(f)$ and  ${\cal L}(g)$ exist.   ${\cal L}(f+g) = {\cal L}(f) 
+ {\cal L}(g)  $

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


[4]~~ 6c.)  Initial conditions will NOT have a significant long term affect on the behavior
of $u(t)$ if $u(t)$ is a solution to the initial value problem:  $2u''(t) + 3u'(t) + 4u(t) =
F(t)$, $u(t_0)  = u_0$, $u'(t_0) = u_0'$. \rightline{T~~~~~~~~~~~F} \w

[4]~~ 6d.)  Initial conditions will NOT have a significant long term affect on the behavior
of $u(t)$ if $u(t)$ is a solution to the initial value problem:  $2u''(t) - 3u'(t) + 4u(t) =
F(t)$. $u(t_0) = u_0$, $u'(t_0) = u_0'$'. \rightline{T~~~~~~~~~~~F} \w


[2]~~ 6e.)  If $3e^{r_1t} + 4e^{r_2t}$ is a solution to the initial value problem 

\centerline{$mu''(t) + \gamma u'(t)  + ku(t) = 0~~u(t_0)  = u_0, ~u'(t_0) = u_0'$}
\vskip -5pt
 and $\psi$ is a solution to the differential equation $mu''(t) +
\gamma u'(t) + ku(t)  = F_o cos(\omega t)$, \break
then $3e^{r_1t} + 4e^{r_2t} + \psi$ is a solution 
to the 
initial value problem 

\centerline{$mu''(t) + \gamma u'(t) + ku(t) = F_o cos(\omega t)~~u(t_0)  = u_0, ~u'(t_0)  =
u_0'$}

\vskip -5pt
\rightline{T~~~~~~~~~~~F}


[3]~~ 6f.)  If $c_1e^{r_1t} + c_2e^{r_2t}$ is a solution to the differential equation 

\centerline{$mu''(t) + \gamma u'(t)  + ku(t) = 0$}
\vskip -5pt
 and $\psi$ is a solution to the differential equation $mu''(t) +
\gamma u'(t) + ku(t)  = F_o cos(\omega t)$, \break
then $c_1e^{r_1t} + c_2e^{r_2t} + \psi$ is a solution 
to the differential equation

\centerline{$mu''(t) + \gamma u'(t) + ku(t) = F_o cos(\omega t)$}

\vskip -5pt
\rightline{T~~~~~~~~~~~F}

\vskip 30pt
[2-4]~~ 7.)  If $u(t)$ is a solution to the initial value problem:  $$mu''(t) +
\gamma u'(t) + ku(t)  = F_o cos(\omega t), ~m, ~\gamma, ~k \geq 0, ~~u(t_0)  = u_0, 
~~u'(t_0)
= u_0'$$ and $lim_{t \rightarrow \infty} |u(t)| = +\infty$, can you say anything about the
values of $m$, $\gamma$, $k$, $F_o$, and $\omega$.





\end
\w
[4]~~ 6d.) If $\psi$ is a solution  to the initial value problem:  $mu''(t) + \gamma u'(t) + 
ku(t) = F(t)$

[3]~~ Eii.)~~ Suppose $c_1cos(3t) + c_2sin3t$ is the solution to the homogeneous equation $mu'' + \gamma u' + ku = 0$, then
resonance occurs if an external force of $F_0cos (\omega t) = 3cos(4t) $ is applied.

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



Suppose the solution to $mu''(t) + \gamma u'(t) + ku(t) = F(t)$.


[2]~~	2a.)  Suppose ${\cal L}(f)$ and  ${\cal L}(g)$ exist.  ${\cal L}(cf) = c{\cal L}(f)  $

[2]~~	2c.)  Suppose ${\cal L}(f)$ and  ${\cal L}(g)$ exist.  ${\cal L}^{-1} (cf) = c{\cal L}^{-1}(f) $  

\rightline{T~~~~~~~~~~~F}
\w
[2]~~	2d.)  Suppose ${\cal L}(f)$ and  ${\cal L}(g)$ exist.   ${\cal L}^{-1}(fg) = {\cal L}^{-1}(f) {\cal L}^{-1}(g)  $

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

[2]~~	2e.)  The delta function is a function.

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

\eject

[4 x 12 = 48] ~~ 4.)  Choose  4 of the following.  Clearly indicate which problems you choose.  You may do all the problems 
for possible partial credit.

\eject 

D.)  Transform $4u'''(t) + u'(t) - 2u(t) = sin(t)$ into a system of first order differential equations. \vskip 3.4in
\vfill \vfill \vfill \vfill \centerline{Answer  $\underline{\hskip 5in}$}

E.)  The following True/False problems refer to the equation $mu'' + \gamma u' + ku = F_0cos (\omega t)$.  Circle T for true and F for false.

[3]~~ Ei.)~~ Suppose $c_1cos(3t) + c_2sin3t$ is the solution to the homogeneous equation $mu'' + \gamma u' + ku = 0$, then
resonance occurs if an external force of $F_0cos (\omega t) = 4cos(3t)$  is applied.

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


[1]~~ Eiii.)~~ Suppose $c_1e^{-3t} + c_2te^{-3t}$ is the solution to the homogeneous equation $mu'' +
\gamma u' + ku = 0$, then resonance occurs if an external force of $F_0cos (\omega t) = 4cos(3t)$ is applied.


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

[1]~~ Eiv.)~~ Suppose $c_1e^{-3t} + c_2te^{-3t}$  is the solution to the homogeneous equation  $mu'' + 
\gamma u' + ku = 0$,  then resonance occurs if an external force of $F_0cos (\omega t) = 3cos(4t) $ is applied.

\rightline{T~~~~~~~~~~~F}
\w
[4]~~ Ev.)~~ If resonance does occur, then $lim_{t \rightarrow \infty} u(t) = 0$


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


\end