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

The solution for y^{'''}-y^{''}-y^'+y=(1+e^{-x}) is y=c_{1}e^{-x}+c_{2}e^x+c_{3}xe^x+1+(e^{-x}x)/4

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

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

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} x e^{x} + 1 + \frac{e ^{- x} x}{4}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} x e^{x} + 1 + \frac{e ^{- x} x}{4}

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} x e^{x} + 1 + \frac{e ^{- x} x}{4}

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

(D^{2} + D + 1) y = e^{x} sin (x)

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

y^{^{''}} + 9 y = 5 sin (3 x), y (0) = 2, y^{^{'}} (0) = 0

y^{^{''}} + 25 y = \frac{1}{100} cos (5 x)

What is the solution for y^{'''}-y^{''}-y^'+y=(1+e^{-x}) ?
The solution for y^{'''}-y^{''}-y^'+y=(1+e^{-x}) is y=c_{1}e^{-x}+c_{2}e^x+c_{3}xe^x+1+(e^{-x}x)/4