What is the solution for y^{''}-2y^'+y=4x^2-6+x^{-1}e^x ?

The solution for y^{''}-2y^'+y=4x^2-6+x^{-1}e^x is y=c_{1}e^x+c_{2}xe^x+4x^2+16x-e^xx+e^xxln(x)+18

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y^{''}-2y^'+y=4x^2-6+x^{-1}e^x

y^{^{''}} - 2 y^{^{'}} + y = 4 x^{2} - 6 + x^{- 1} e^{x}

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} + 16 x - e^{x} x + e^{x} x ln (x) + 18

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} + 16 x - e^{x} x + e^{x} x ln (x) + 18

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} + 16 x - e^{x} x + e^{x} x ln (x) + 18

2 y^{^{''}} - 4 y^{^{'}} - 6 y = 3 e^{2 x}

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

y^{^{''}} + 16 y = 7 sec (4 x)

y^{^{''}} + y^{^{'}} + 2 y = - 25 x e^{x}, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} + 13 = 0

What is the solution for y^{''}-2y^'+y=4x^2-6+x^{-1}e^x ?
The solution for y^{''}-2y^'+y=4x^2-6+x^{-1}e^x is y=c_{1}e^x+c_{2}xe^x+4x^2+16x-e^xx+e^xxln(x)+18