What is the solution for (d^2y)/(dx^2)-(dy)/(dx)-6y=-e^x+12x ?

The solution for (d^2y)/(dx^2)-(dy)/(dx)-6y=-e^x+12x is y=c_{1}e^{3x}+c_{2}e^{-2x}+1/6 e^x-2x+1/3

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(d^2y)/(dx^2)-(dy)/(dx)-6y=-e^x+12x

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - 6 y = - e^{x} + 12 x

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} + \frac{1}{6} e^{x} - 2 x + \frac{1}{3}

Solution steps

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - 6 y = - e^{x} + 12 x

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} + \frac{1}{6} e^{x} - 2 x + \frac{1}{3}

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} + \frac{1}{6} e^{x} - 2 x + \frac{1}{3}

What is the solution for (d^2y)/(dx^2)-(dy)/(dx)-6y=-e^x+12x ?
The solution for (d^2y)/(dx^2)-(dy)/(dx)-6y=-e^x+12x is y=c_{1}e^{3x}+c_{2}e^{-2x}+1/6 e^x-2x+1/3