What is the solution for y^{''}-4y^'+9y=2x-3 ?

The solution for y^{''}-4y^'+9y=2x-3 is y=e^{2x}(c_{1}cos(sqrt(5)x)+c_{2}sin(sqrt(5)x))+(2x)/9-19/81

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y^{''}-4y^'+9y=2x-3

y^{^{''}} - 4 y^{^{'}} + 9 y = 2 x - 3

y = e^{2 x} (c_{1} cos (5 x) + c_{2} sin (5 x)) + \frac{2 x}{9} - \frac{19}{81}

Solution steps

y^{''} (x) - 4 y^{'} (x) + 9 y = 2 x - 3

Solve linear ODE:

y = e^{2 x} (c_{1} cos (5 x) + c_{2} sin (5 x)) + \frac{2 x}{9} - \frac{19}{81}

y = e^{2 x} (c_{1} cos (5 x) + c_{2} sin (5 x)) + \frac{2 x}{9} - \frac{19}{81}

y^{^{''}} + 2 y^{^{'}} + y = sin (x) + 8 cos (2 x)

(D^{4} - 2 D^{2} + 1) y = cos (x)

y^{^{''}} + 5 y^{^{'}} + 4 y = 2 x e^{2 x}

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

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

What is the solution for y^{''}-4y^'+9y=2x-3 ?
The solution for y^{''}-4y^'+9y=2x-3 is y=e^{2x}(c_{1}cos(sqrt(5)x)+c_{2}sin(sqrt(5)x))+(2x)/9-19/81