What is the solution for (d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x ?

The solution for (d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x is y=c_{1}e^{2x}+c_{2}xe^{2x}+e^x

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(d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 4 y = e^{x}

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 4 y = e^{x}

Solve linear ODE:

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

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

y^{^{''''}} - 4 y^{^{''}} = x^{2}

y^{^{''}} + 2 y^{^{'}} + 5 y = 12 e^{- t} cos (2 t)

y^{^{''}} - 4 y^{^{'}} + 8 y = x^{3}

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

2 y^{^{''}} - 7 y^{^{'}} - 4 y = 7 - 2 x

What is the solution for (d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x ?
The solution for (d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x is y=c_{1}e^{2x}+c_{2}xe^{2x}+e^x