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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

2 y^{^{''}} + 6 y^{^{'}} = 5

y^{^{''}} - 5 y^{^{'}} + 6 y = 4 x e^{2 x}, y^{1} (0) = e^{2 x}

y^{^{''}} - 5 y^{^{'}} + 6 y = t, y (0) = 0, y^{^{'}} (0) = 1

y^{^{'''}} + 18 y^{^{''}} + 108 y^{^{'}} + 216 y = e^{- 6 x}

2 \cdot y + 2 \cdot y^{^{'}} + y^{^{''}} = e^{- x} cos (x)

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