10 Ordinary Differential Equations II

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Solve using class procedures
   
   $$y'' + 2 y' + 3 y = 0 \qquad y(0) = 6 \qquad y'(0) = 2$$
   
   

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

3. Solve using variation of parameters,
   
   $$2 y'' - 4y' -6 y = 4 \sin^2(x)$$
   
   

4. 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.

5. Solve
   
   $$y' - 9 y = t \qquad y(0) = 5$$
   
   
   That would of course be quick using undetermined coefficients, or solving as a first order linear equation. 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 3 equations in 3 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.