What is the solution for y^{''}-8y^'-9y=2e^{-x}-9x ?

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

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y^{''}-8y^'-9y=2e^{-x}-9x

y^{^{''}} - 8 y^{^{'}} - 9 y = 2 e^{- x} - 9 x

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

Solution steps

y^{''} (x) - 8 y^{'} (x) - 9 y = 2 e^{- x} - 9 x

Solve linear ODE:

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

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

y^{^{''}} - 5 y^{^{'}} + 4 y = x, y^{1} (0) = e^{x}

y^{^{''}} - 5 y^{^{'}} + 6 y = 2 e^{- x}, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} + 5 y = e^{t}

y^{^{'''}} - y^{^{''}} - y^{^{'}} + y = - 2 sin (x) + 3 e^{x}

y^{^{''}} + y = 1, y^{^{'}} (0) = - 1

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