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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + 81 y = 2 sin (9 t)

y^{^{''}} - 4 y = 12 x, y (0) = 4, y^{^{'}} (0) = 1

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

y^{^{''}} - 2 y^{^{'}} - 15 y = 384 e^{- t}

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