Problems

01

Superposition and sums of forcing terms

Find a complete set of solutions to the ODE given by $y^{\prime \prime }+9y=2\sin 3t+4\sin t-26e^{-2t}$.

02

Discovering a second solution

Find a formula (maybe involving integrals) for the complete family of solutions to the following ODE:

$$y^{\prime \prime }-tf(t)y^{\prime }+f(t)y=0.$$

Your answer should be written in terms of the unknown function $f(t)$. Hint: first guess an easy solution.

03

Variation of parameters

Using variation of parameters, find particular solutions to the following ODEs:

• (a) $y^{\prime \prime }+2y^{\prime }+y=e^{-t}\ln t$
• (b) $y^{\prime \prime }+y=\sec (t)\tan (t)$

04

Fourier solution

Using variation of parameters, show that the solution to the IVP

$$y^{\prime \prime }+y=g(t),y(t_0)=0,y^{\prime }(t_0)=0$$

is given by $y(t)={\int }_{t_0}^t\sin (t-s)g(s)ds$.