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

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

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

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

y = c_{1} e^{x} + c_{2} x e^{x} - e^{x} ln (x) - e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} - e^{x} ln (x) - e^{x}

y = c_{1} e^{x} + c_{2} x e^{x} - e^{x} ln (x) - e^{x}

x^{^{''}} + 2 x^{^{'}} + 5 x = sin (2 t)

y^{^{''}} - 9 y^{^{'}} = 3 x^{2} - 5 sin (3 x)

y^{^{''}} - y^{^{'}} - 2 y = 3 e^{- x}

4 y^{^{''}} + 16035 y = 4 cos (8 t)

y^{^{''}} + 3 y^{^{'}} + 2 y = 12 e^{t}

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