Pregunta: Solve the initial value problem y′′+2y′+2y=8e−tsint;y(0)=0,y′(0)=1.

Ans We have to solve the initial value problem $\mathrm{y}{}^{″}+2{\mathrm{y}}^{\prime }+2\mathrm{y}=8{\mathrm{e}}^{-\mathrm{t}}\sin \mathrm{t};$$\mathrm{y}\left(0\right)=0,{\mathrm{y}}^{\prime }\left(0\right)=1.$

$({\mathrm{D}}^2+2\mathrm{D}+2)\mathrm{y}=8{\mathrm{e}}^{-\mathrm{t}}\sin \mathrm{t}$

For the homogeneous part :