What is the solution for y^{''}-6y^'+9y=8e^{3x} ?

The solution for y^{''}-6y^'+9y=8e^{3x} is y=c_{1}e^{3x}+c_{2}xe^{3x}+4x^2e^{3x}

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y^{''}-6y^'+9y=8e^{3x}

y^{^{''}} - 6 y^{^{'}} + 9 y = 8 e^{3 x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + 4 x^{2} e^{3 x}

Solution steps

y^{''} (x) - 6 y^{'} (x) + 9 y = 8 e^{3 x}

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + 4 x^{2} e^{3 x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x} + 4 x^{2} e^{3 x}

y^{^{''}} + 4 y = 8 tan^{2} (2 x)

y^{^{''}} + 4 y^{^{'}} = 8

y^{^{'''}} - 4 y^{^{''}} + 5 y^{^{'}} = 40 t - 7 - 6 e^{2 t}

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

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

What is the solution for y^{''}-6y^'+9y=8e^{3x} ?
The solution for y^{''}-6y^'+9y=8e^{3x} is y=c_{1}e^{3x}+c_{2}xe^{3x}+4x^2e^{3x}