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$1_a$.  Find the general solution of the differential equation:
$y''+2y'+3y=0.$
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$1_b$.  Solve the following initial value problem:
$y''+4y=t^2 + 3e^{t}$,  $y(0)=0$, $y'(0)=0$.
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$2$. Find the general solution of the differential equation:
$ty'' + y' = t$,  $t>1$.

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3. Find the general solution of the differential equation:
$y'' - 10y' + 25y = x^{-2} e^{5x}, \, x>0.$

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$4_a$.  If the Wronskian of $f$ and $g$ is 
$W(f,g) = x^{2} \cos{x}$ and if $u=f + g$ and $v=f-g$, find the 
Wronskian of $u$ and $v$.

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$4_b$. Suppose that $y'' - ty = \cos{t}$ and that $y(0)=1$ and $y'(0)=\pi$. 
Find $y'''(0)$.

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5. A mass $m$ stretches a spring $2$ ft. If the mass is set in motion from its 
equilibrium position with a downward velocity of $16$ ft/sec and if there is 
no air resistance or other damping factors, determine the position of the 
mass at any time. What is the period of the motion? (Assume the 
gravitational constant $g=32$ ft/sec$^2$).


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