What is the solution for (d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x*(x-1) ?

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

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(d^2y)/(dx^2)-4(dy)/(dx)+4y=e^x*(x-1)

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 4 y = e^{x} \cdot (x - 1)

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 4 y = e^{x} \cdot (x - 1)

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + y = 0 + e^{2 x}

y^{^{''}} + 2 y = 2, 5 e^{- t} y (0) = 3

y^{^{''}} + 4 y = 5 t e^{t}

y^{^{''}} - 2 y^{^{'}} - 12 y = 2 x + 6

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 e^{- 5 t}

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