What is the solution for y^{''}+3y^'=2t^4+t^2e^{-3t}+sin(3t) ?

The solution for y^{''}+3y^'=2t^4+t^2e^{-3t}+sin(3t) is

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y^{''}+3y^'=2t^4+t^2e^{-3t}+sin(3t)

y^{^{''}} + 3 y^{^{'}} = 2 t^{4} + t^{2} e^{- 3 t} + sin (3 t)

y = c_{1} + c_{2} e^{- 3 t} + \frac{324 t ^{5} - 540 t ^{4} + 720 t ^{3} - 270 e ^{- 3 t} t ^{3} - 270 e ^{- 3 t} t ^{2} - 720 t ^{2} - 180 e ^{- 3 t} t + 480 t - 60 e ^{- 3 t} - 135 cos ( 3 t ) - 135 sin ( 3 t ) - 160}{2430}

Solution steps

y^{''} (t) + 3 y^{'} (t) = 2 t^{4} + t^{2} e^{- 3 t} + sin (3 t)

Solve linear ODE:

y = c_{1} + c_{2} e^{- 3 t} + \frac{324 t ^{5} - 540 t ^{4} + 720 t ^{3} - 270 e ^{- 3 t} t ^{3} - 270 e ^{- 3 t} t ^{2} - 720 t ^{2} - 180 e ^{- 3 t} t + 480 t - 60 e ^{- 3 t} - 135 cos ( 3 t ) - 135 sin ( 3 t ) - 160}{2430}

y = c_{1} + c_{2} e^{- 3 t} + \frac{324 t ^{5} - 540 t ^{4} + 720 t ^{3} - 270 e ^{- 3 t} t ^{3} - 270 e ^{- 3 t} t ^{2} - 720 t ^{2} - 180 e ^{- 3 t} t + 480 t - 60 e ^{- 3 t} - 135 cos ( 3 t ) - 135 sin ( 3 t ) - 160}{2430}

y^{^{'}} = x y^{2} + 2 x y

n = 1 \sum \infty \frac{1}{n ^{\frac{2}{3}}}

\int 6^{(x - 2)} d x

\int \frac{1}{5 - x ^{2} - 4 x} d x

x^{3} \frac{d y}{d x} = y^{2} + 1

What is the solution for y^{''}+3y^'=2t^4+t^2e^{-3t}+sin(3t) ?
The solution for y^{''}+3y^'=2t^4+t^2e^{-3t}+sin(3t) is