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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 3 y = - 48 x^{2} e^{3} x

y^{^{'''}} - y^{^{''}} - 25 y^{^{'}} + 25 y = 2 - e^{x} + e^{5 x}

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

y^{^{''}} + 2 y^{^{'}} - 3 y = 16 e^{- 3 t}

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

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