What is the solution for (d^2y)/(dx^2)-y=xe^{2x} ?

The solution for (d^2y)/(dx^2)-y=xe^{2x} is y=c_{1}e^x+c_{2}e^{-x}+(e^{2x}x)/3-4/9 e^{2x}

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(d^2y)/(dx^2)-y=xe^{2x}

\frac{d ^{2} y}{d x ^{2}} - y = x e^{2 x}

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{2 x} x}{3} - \frac{4}{9} e^{2 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} - y = x e^{2 x}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{2 x} x}{3} - \frac{4}{9} e^{2 x}

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{2 x} x}{3} - \frac{4}{9} e^{2 x}

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

y^{^{''}} + 5 y^{^{'}} + 4 y = 8 x

y^{^{''}} + 5 y^{^{'}} + 6 y = e^{2 x} + 1

y^{^{''}} - y^{^{'}} - 2 y = 5 e^{4 x} + 6 x

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = x + e^{- 3 x}

What is the solution for (d^2y)/(dx^2)-y=xe^{2x} ?
The solution for (d^2y)/(dx^2)-y=xe^{2x} is y=c_{1}e^x+c_{2}e^{-x}+(e^{2x}x)/3-4/9 e^{2x}