What is the solution for y^{''}+3y^'=2x^4+x^2e^{-3x}+sin(3x) ?

The solution for y^{''}+3y^'=2x^4+x^2e^{-3x}+sin(3x) is

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y^{''}+3y^'=2x^4+x^2e^{-3x}+sin(3x)

y^{^{''}} + 3 y^{^{'}} = 2 x^{4} + x^{2} e^{- 3 x} + sin (3 x)

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

Solution steps

y^{''} (x) + 3 y^{'} (x) = 2 x^{4} + x^{2} e^{- 3 x} + sin (3 x)

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + 5 y = 1 + t, y (0) = 0, y^{^{'}} (0) = 4

x^{^{''}} + 4 x^{^{'}} + 5 x = 13 sin (2 t)

y^{^{''}} - 3 y^{^{'}} + 4 y = 16 x - 12 e^{2 x}

y^{^{''}} - 7 y^{^{'}} = (x - 1)^{2}

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

What is the solution for y^{''}+3y^'=2x^4+x^2e^{-3x}+sin(3x) ?
The solution for y^{''}+3y^'=2x^4+x^2e^{-3x}+sin(3x) is