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

The solution for y^{''}+2y^'+y=x^2e-x is y=c_{1}e^{-x}+c_{2}xe^{-x}+ex^2+(-1-4e)x+6e+2

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

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

y = c_{1} e^{- x} + c_{2} x e^{- x} + e x^{2} + (- 1 - 4 e) x + 6 e + 2

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} + e x^{2} + (- 1 - 4 e) x + 6 e + 2

y = c_{1} e^{- x} + c_{2} x e^{- x} + e x^{2} + (- 1 - 4 e) x + 6 e + 2

What is the solution for y^{''}+2y^'+y=x^2e-x ?
The solution for y^{''}+2y^'+y=x^2e-x is y=c_{1}e^{-x}+c_{2}xe^{-x}+ex^2+(-1-4e)x+6e+2