What is the solution for y^{''}-4y^'+4y=x^{-1}e^{2x} ?

The solution for y^{''}-4y^'+4y=x^{-1}e^{2x} is y=c_{1}e^{2x}+c_{2}xe^{2x}-e^{2x}x+e^{2x}xln(x)

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

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

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

Solution steps

y^{''} (x) - 4 y^{'} (x) + 4 y = x^{- 1} e^{2 x}

Solve linear ODE:

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

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

16 y^{^{''}} - 24 y^{^{'}} + 9 y = 5 - 3 x

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

y^{^{''}} + 4 y^{^{'}} + 40 y = 50 sin (6 t)

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

y^{^{''}} + 4 y = sin (3 t), y (0) = 0, y^{^{'}} (0) = 0

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