What is the solution for y^{''}-4y^'-5y=4x^2e^{2x} ?

The solution for y^{''}-4y^'-5y=4x^2e^{2x} is y=c_{1}e^{5x}+c_{2}e^{-x}-(4e^{2x}x^2)/9-8/81 e^{2x}

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y^{''}-4y^'-5y=4x^2e^{2x}

y^{^{''}} - 4 y^{^{'}} - 5 y = 4 x^{2} e^{2 x}

y = c_{1} e^{5 x} + c_{2} e^{- x} - \frac{4 e ^{2 x} x ^{2}}{9} - \frac{8}{81} e^{2 x}

Solution steps

y^{''} (x) - 4 y^{'} (x) - 5 y = 4 x^{2} e^{2 x}

Solve linear ODE:

y = c_{1} e^{5 x} + c_{2} e^{- x} - \frac{4 e ^{2 x} x ^{2}}{9} - \frac{8}{81} e^{2 x}

y = c_{1} e^{5 x} + c_{2} e^{- x} - \frac{4 e ^{2 x} x ^{2}}{9} - \frac{8}{81} e^{2 x}

What is the solution for y^{''}-4y^'-5y=4x^2e^{2x} ?
The solution for y^{''}-4y^'-5y=4x^2e^{2x} is y=c_{1}e^{5x}+c_{2}e^{-x}-(4e^{2x}x^2)/9-8/81 e^{2x}