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

The solution for (d^2y)/(dx^2)-y=x*e^{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=x*e^{2x}

\frac{d ^{2} y}{d x ^{2}} - y = x \cdot 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 \cdot 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^{^{''}} + 4 y^{^{'}} + 4 y = 5 x + 4 - e^{5 x}

y^{^{''}} - 6 y^{^{'}} + 5 y = 34 sin (3 t)

y^{^{''}} - 9 y^{^{'}} + 2 y = t e^{t}

y^{^{''}} + 4 y = 8 sin (2 t) + 5 e^{t}

y^{^{''}} - 8 y^{^{'}} + 16 y = 45 e^{7 t}

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