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

The solution for y^{''}+y=e^{-x}+2x^2+3 is y=c_{1}cos(x)+c_{2}sin(x)+1/2 e^{-x}+2x^2-1

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

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

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{2} e^{- x} + 2 x^{2} - 1

Solution steps

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

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{2} e^{- x} + 2 x^{2} - 1

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{2} e^{- x} + 2 x^{2} - 1

y^{^{''}} + 4 y = 2 csc^{2} (2 x)

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

y^{^{''}} - 2 y^{^{'}} = 3 x^{2} + 1

(D^{2} + 100) (D^{2} - 100) (D + 10)^{4} x = 100

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

What is the solution for y^{''}+y=e^{-x}+2x^2+3 ?
The solution for y^{''}+y=e^{-x}+2x^2+3 is y=c_{1}cos(x)+c_{2}sin(x)+1/2 e^{-x}+2x^2-1