What is the solution for y^{''}-2y^'+y=(12e^x)/(x^3) ?

The solution for y^{''}-2y^'+y=(12e^x)/(x^3) is y=c_{1}e^x+c_{2}xe^x+(6e^x)/x

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y^{''}-2y^'+y=(12e^x)/(x^3)

y^{^{''}} - 2 y^{^{'}} + y = \frac{12 e ^{x}}{x ^{3}}

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

Solution steps

y^{''} (x) - 2 y^{'} (x) + y = \frac{12 e ^{x}}{x ^{3}}

Solve linear ODE:

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

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

(D^{4} + D^{2}) y = x^{2} + 2 x + 4

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - 12 y = (x + 1) e^{x}

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

y^{^{''}} - 12 y^{^{'}} + 36 y = 12 e^{6 x}

y^{^{''}} - 4 y^{^{'}} = e^{x}

What is the solution for y^{''}-2y^'+y=(12e^x)/(x^3) ?
The solution for y^{''}-2y^'+y=(12e^x)/(x^3) is y=c_{1}e^x+c_{2}xe^x+(6e^x)/x