Tutorial Week 4

Worksheet Questions

This worksheet is not to be handed in. It should be considered a review/practice sheet for the previous week's material in preparation for the midterm tomorrow. For review/practice of earlier material, look at the old midterms posted on the course site ("solutions" on the menu) and previous worksheets.

1. The differential equation $y'' + 2\alpha^2 y' + (\alpha^4-\alpha^2+2\alpha) y = 0$ depends on the parameter $\alpha$ so that its solutions may be qualitatively different for different values of $\alpha$.
   1. For what values of $\alpha$ does the characteristic equation of the system have real distinct roots?
   2. For what values of $\alpha$ does the characteristic equation of the system have a real repeated root?
   3. For what values of $\alpha$ does the characteristic equation of the system have complex roots?
   4. Assuming complex roots, what is the general solution?
   5. At what time will the amplitude have decayed to $e^{-1}$ of its original value?
2. For each of the following differential equations, write the general form of its particular solution.
   1. $y''-25y'=2e^{8x}+3$
   2. $y''-y=x\cos(x)$
   3. $y''+y=x\cos(x)$
   4. $y''-2y'+y=e^{x}$
3. Find the general solution to the equation $y''+9y=\sin(3t)$.

Tutorial Week 4 Solutions