Exercises

01

Variation of parameters: double repeated root, inhomogeneous

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

02

Variation of parameters: second solution from one obvious.

Find a complete set of solutions to the ODE given by $y^{\prime \prime }-tf(t)y^{\prime }+f(t)y=0$.

03

Solving inhomogeneous linear ODE

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

04

Solving inhomogeneous linear ODE

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

05

Solving inhomogeneous linear ODE

Find a complete set of solutions to the ODE given by $y^{\prime \prime }+k^{2}y=\sin (\omega t)$, where $k,\omega >0$ are constants.

06

Superposition and sums of forcing terms

Find a complete set of solutions to the ODE given by $y^{\prime \prime }+4y=6\cos (t)+8t^2$.

(Hint: add two particular solutions together, each solving an inhomogeneous equation corresponding to a single one of the forcing terms.)