What is the solution for y^{''}+2y^'+2y=x^2-3xe^{-2x}cos(5x) ?

The solution for y^{''}+2y^'+2y=x^2-3xe^{-2x}cos(5x) is

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y^{''}+2y^'+2y=x^2-3xe^{-2x}cos(5x)

y^{^{''}} + 2 y^{^{'}} + 2 y = x^{2} - 3 x e^{- 2 x} cos (5 x)

y = e^{- x} (c_{1} cos (x) + c_{2} sin (x)) + e^{- x} cos (x) \cdot \int - e^{x} sin (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x + e^{- x} sin (x) \cdot \int e^{x} cos (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x

Solution steps

y^{''} (x) + 2 y^{'} (x) + 2 y = x^{2} - 3 x e^{- 2 x} cos (5 x)

Solve linear ODE:

y = e^{- x} (c_{1} cos (x) + c_{2} sin (x)) + e^{- x} cos (x) \cdot \int - e^{x} sin (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x + e^{- x} sin (x) \cdot \int e^{x} cos (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x

y = e^{- x} (c_{1} cos (x) + c_{2} sin (x)) + e^{- x} cos (x) \cdot \int - e^{x} sin (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x + e^{- x} sin (x) \cdot \int e^{x} cos (x) (x^{2} - 3 e^{- 2 x} x cos (5 x)) d x

What is the solution for y^{''}+2y^'+2y=x^2-3xe^{-2x}cos(5x) ?
The solution for y^{''}+2y^'+2y=x^2-3xe^{-2x}cos(5x) is