What is the solution for y^{''}+106y^'+26y=2t^3e^{-5t}sin(t) ?

The solution for y^{''}+106y^'+26y=2t^3e^{-5t}sin(t) is

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y^{''}+106y^'+26y=2t^3e^{-5t}sin(t)

y^{^{''}} + 106 y^{^{'}} + 26 y = 2 t^{3} e^{- 5 t} sin (t)

y = c_{1} e^{(- 53 + 1123) t} + c_{2} e^{(- 53 - 1123) t} + e^{- 53 t + 1123 t} \cdot \int \frac{23 e ^{48 t - 1123 t} t ^{3} sin ( t )}{253} d t + e^{- 53 t - 1123 t} \cdot \int - \frac{23 e ^{48 t + 1123 t} t ^{3} sin ( t )}{253} d t

Solution steps

y^{''} (t) + 106 y^{'} (t) + 26 y = 2 t^{3} e^{- 5 t} sin (t)

Solve linear ODE:

y = c_{1} e^{(- 53 + 1123) t} + c_{2} e^{(- 53 - 1123) t} + e^{- 53 t + 1123 t} \cdot \int \frac{23 e ^{48 t - 1123 t} t ^{3} sin ( t )}{253} d t + e^{- 53 t - 1123 t} \cdot \int - \frac{23 e ^{48 t + 1123 t} t ^{3} sin ( t )}{253} d t

y = c_{1} e^{(- 53 + 1123) t} + c_{2} e^{(- 53 - 1123) t} + e^{- 53 t + 1123 t} \cdot \int \frac{23 e ^{48 t - 1123 t} t ^{3} sin ( t )}{253} d t + e^{- 53 t - 1123 t} \cdot \int - \frac{23 e ^{48 t + 1123 t} t ^{3} sin ( t )}{253} d t

y^{^{''}} - 7 y^{^{'}} = 5 x - 3

y^{^{''}} - 2 y^{^{'}} + 50 y = e^{x} cos (7 x)

\frac{d ^{2} y}{d x ^{2}} - y = e^{x}

y^{^{''}} + 25 y^{^{'}} = - 24 t e^{- t}

y^{^{''}} + 4 y^{^{'}} + y = (\frac{e ^{- \frac{x}{2}}}{x ^{2}})

What is the solution for y^{''}+106y^'+26y=2t^3e^{-5t}sin(t) ?
The solution for y^{''}+106y^'+26y=2t^3e^{-5t}sin(t) is