What is the solution for (d^2+9)y=5e^x-162x ?

The solution for (d^2+9)y=5e^x-162x is y=c_{1}cos(3x)+c_{2}sin(3x)+1/2 e^x-18x

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(d^2+9)y=5e^x-162x

(d^{2} + 9) y = 5 e^{x} - 162 x

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{2} e^{x} - 18 x

Solution steps

(d^{2} + 9) y = 5 e^{x} - 162 x

Solve linear ODE:

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{2} e^{x} - 18 x

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{2} e^{x} - 18 x

y^{^{''}} - 3 y^{^{'}} - 9 y = 4 cos (2 x) - 5 e^{- x}

y^{^{''}} - 2 y^{^{'}} = x e^{3 x}

y^{^{'''}} - 3 y^{^{''}} + 4 y = e^{2 x}

y^{^{''}} + 9 y = 3 cos (3 x)

y^{^{''}} + 4 y^{^{'}} + 4 y = 5 e^{- 2 t}

What is the solution for (d^2+9)y=5e^x-162x ?
The solution for (d^2+9)y=5e^x-162x is y=c_{1}cos(3x)+c_{2}sin(3x)+1/2 e^x-18x