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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - y^{^{'}} = 4 x e^{x}, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} - 6 y^{^{'}} + 9 y = e^{3 x}, y (0) = - 1, y^{^{'}} (0) = 1

y^{^{''}} + 10 y^{^{'}} + 61 y = 5 cos (7 t)

(D^{4} - 1) y = 7 x^{2}

y^{^{''}} - 2 y^{^{'}} - 3 y = 4 e^{5 x}

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