What is the solution for y^{''}-6y+9y=e^{3x} ?

The solution for y^{''}-6y+9y=e^{3x} is y=c_{1}cos(sqrt(3)x)+c_{2}sin(sqrt(3)x)+1/12 e^{3x}

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y^{''}-6y+9y=e^{3x}

y^{^{''}} - 6 y + 9 y = e^{3 x}

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{12} e^{3 x}

Solution steps

y^{''} (x) - 6 y + 9 y = e^{3 x}

Solve linear ODE:

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{12} e^{3 x}

y = c_{1} cos (3 x) + c_{2} sin (3 x) + \frac{1}{12} e^{3 x}

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

3 y^{^{''}} - 3 y^{^{'}} + y = - 18 + 31 x - 11 x^{2} + x^{3}

y^{^{''}} + \frac{2}{5} y^{^{'}} + \frac{1}{50} y = 100

y^{^{''}} - y^{^{'}} - 2 y = e^{- t}, y (0) = - 1, y^{^{'}} (0) = 2

2 y^{^{''}} + 9 y^{^{'}} - y = 22

What is the solution for y^{''}-6y+9y=e^{3x} ?
The solution for y^{''}-6y+9y=e^{3x} is y=c_{1}cos(sqrt(3)x)+c_{2}sin(sqrt(3)x)+1/12 e^{3x}