What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=(e^x)/(x^3) ?

The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=(e^x)/(x^3) is y=c_{1}e^x+c_{2}xe^x+(e^x)/(2x)

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(d^2y)/(dx^2)-2(dy)/(dx)+y=(e^x)/(x^3)

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

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

Solution steps

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

Solve linear ODE:

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

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

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

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

y^{^{''}} + 2 y^{^{'}} + y = e^{3 x}, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} + 3 y = 9 x^{2}

y^{^{''}} + 6 y^{^{'}} - 7 y = 21 x^{2} - 8 x - 30

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