$10$

Use Laplace Transforms to solve the initial value promem

$y^{\text{'}\text{'}}+4y=8t,y(0)=3,y^{\text{'}}(0)=0$

$y(t)=2t+2cos2t+2sin2t,y(t)=t+2cos2t-2sin2t$

$y(t)=2t+3cos2t-sin2t,y(t)=t-3cos2t+sin2t$

$y(t)=t-2cos2t-sin2t$