10 Ordinary Differential Equations II

1. Solve using undetermined coefficients:
   
   $$y'' - 2y' + y = 3x + 25 \sin(3x) + 2 e^x \qquad y(0)=1 \quad y'(0)=2$$
   
   

2. Find the Laplace transform $\widehat u$ of
   
   $$u = 1 - 4 t + 2 t^2 e^{-3t}$$
   
   
   You may only use the brief Laplace transform table handed out in class. Everything else must be derived. Do not use convolution.

3. Solve
   
   $$y'' + 9 y = t^2 \qquad y(0) = y'(0)= 0$$
   
   
   That would of course be quick using undetermined coefficients. Unfortunately, you must use Laplace transforms. You may only use the brief Laplace transform table handed out in class. Everything else must be derived. Do not use convolution. In solving the system of 5 equations in 5 unknowns of the partial fraction expansion, you may mess around; this is no longer linear algebra. However, you must substitute your solution into the original ODE and ICs and go back to fix any problem there may be.

4. Resonant forcing of an undamped spring-mass system over some time period $T$ that spans a large number of periods can introduce large-amplitude vibrations. To study the problem, consider the example
   
   $$m \ddot x + k x = F(t) \qquad x(0) = \dot x(0) = 0$$
   
   
   where the mass, spring constant, and applied force are given by
   
   $$m=1 \qquad k=4 \qquad F(t)=\cos(2t)\mbox{ if } t<T \qquad F(t)= 0 \mbox{ if } t>T$$
   
   
   Solve using the Laplace transform method. Note: from S8 and S11 you can see that
   
   $$\sin(\omega t) - \omega t \cos(\omega t) \Longleftrightarrow \frac{2\omega^3}{(s^2+\omega^2)^2}$$
   
   
   Clean up your answer. I find that beyond time $t=T$, the amplitude stays constant at
   
   $${\textstyle\frac{1}{4}} T \sqrt{1 + 2\cos(2T)\frac{\sin(2T)}{2T} + \left(\frac{\sin(2T)}{2T}\right)^2}$$
   
   
   which is approximately proportional to $T$ for large $T$. Do your results agree?