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

The solution for y^{''}-2y^'+y=xe^x+e^{-x} is y=c_{1}e^x+c_{2}xe^x+(e^xx^3)/6+1/4 e^{-x}

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

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

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{3}}{6} + \frac{1}{4} e^{- x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{3}}{6} + \frac{1}{4} e^{- x}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{3}}{6} + \frac{1}{4} e^{- x}

y^{^{''}} - y^{^{'}} = x e [x], y (0) = 2, y^{^{'}} (0) = 1

y^{^{''}} + y = 2 sin (2 t)

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

y^{^{''}} - y^{^{'}} = t^{2}, y (0) = 0, y^{^{'}} (0) = 0

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

What is the solution for y^{''}-2y^'+y=xe^x+e^{-x} ?
The solution for y^{''}-2y^'+y=xe^x+e^{-x} is y=c_{1}e^x+c_{2}xe^x+(e^xx^3)/6+1/4 e^{-x}