Chapter D2.1, Problem 67E

a.

To determine

To verify: The function $y=e^{-3t}\left(c_1\sin 4t+c_2\cos 4t\right)+e^{-t}$ is a solution of the differential equation $y^{″}+6y^{\prime }+25y=20e^{-t}$.

b.

To determine

To find: The solution of the initial value problem $y^{″}+6y^{\prime }+25y=20e^{-t};y\left(0\right)=2,y^{\prime }\left(0\right)=0$.

c.

To determine

To sketch: The function $y=e^{-3t}\left(\sin 4t+\cos 4t\right)+e^{-t}$.