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

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

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

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

y = c_{1} e^{2 x} + c_{2} e^{- x} + e^{2 x} x + \frac{x ^{2} - x}{2} + \frac{3}{4}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} + e^{2 x} x + \frac{x ^{2} - x}{2} + \frac{3}{4}

y = c_{1} e^{2 x} + c_{2} e^{- x} + e^{2 x} x + \frac{x ^{2} - x}{2} + \frac{3}{4}

y^{^{''}} + 3 y^{^{'}} = 3

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

y^{^{''}} + 4 y^{^{'}} + 8 y = sin (x)

y^{^{''}} + 2 y^{^{'}} + 5 y = 12 cos (2 x) + 3 sin (2 x)

y^{^{''}} + 4 y = 2 x, y (0) = 1, y^{^{'}} (0) = 2

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