What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)-3y=2e^{3x} ?

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

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(d^2y)/(dx^2)-2(dy)/(dx)-3y=2e^{3x}

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

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

Solution steps

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

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 32 y = (x + 1) e^{x}

2 y^{^{''}} + y^{^{'}} - y = e^{3 t}, y (0) = 2, y^{^{'}} (0) = 0

y^{^{''}} = y + e^{2 x} \cdot e^{e^{x}}

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

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

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