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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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