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

The solution for y^{''}-y^'-2y=-2x+4x^2 is y=c_{1}e^{2x}+c_{2}e^{-x}-2x^2+3x-7/2

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y^{''}-y^'-2y=-2x+4x^2

y^{^{''}} - y^{^{'}} - 2 y = - 2 x + 4 x^{2}

y = c_{1} e^{2 x} + c_{2} e^{- x} - 2 x^{2} + 3 x - \frac{7}{2}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} - 2 x^{2} + 3 x - \frac{7}{2}

y = c_{1} e^{2 x} + c_{2} e^{- x} - 2 x^{2} + 3 x - \frac{7}{2}

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

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

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

y^{^{''}} + 8 y^{^{'}} + 16 y = \frac{2 e ^{- 4 x}}{x ^{2}}

y^{^{''}} - 2 y^{^{'}} + y = t^{- 6} e^{t}

What is the solution for y^{''}-y^'-2y=-2x+4x^2 ?
The solution for y^{''}-y^'-2y=-2x+4x^2 is y=c_{1}e^{2x}+c_{2}e^{-x}-2x^2+3x-7/2