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

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

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y^{''}-2y^'+y=6xe^x+4e^x

y^{^{''}} - 2 y^{^{'}} + y = 6 x e^{x} + 4 e^{x}

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

Solution steps

y^{''} (x) - 2 y^{'} (x) + y = 6 x e^{x} + 4 e^{x}

Solve linear ODE:

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

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

y^{^{''}} + 4 y^{^{'}} + 4 y = 3 x + 4

(D^{2} + D - 12) y = x

y^{^{''}} - 8 y^{^{'}} + 16 y = (x - 1) e^{4 x}

y^{^{''}} + y = t - (t - 4) H (t - 2)

y^{^{''}} - 5 y^{^{'}} - 14 y = e^{x} (x + 1)

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