What is the solution for y^{''}-2y^'+y=2xe^{2x}+6e^x ?

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

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y^{''}-2y^'+y=2xe^{2x}+6e^x

y^{^{''}} - 2 y^{^{'}} + y = 2 x e^{2 x} + 6 e^{x}

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

Solution steps

y^{''} (x) - 2 y^{'} (x) + y = 2 x e^{2 x} + 6 e^{x}

Solve linear ODE:

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

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

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