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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + 7 y^{^{'}} - 12 = e^{3 x} - 2 e^{x} - 5 x

y^{^{''}} + 2 y^{^{'}} + 2 y = 1 + cos (2 t)

y^{^{''}} + 4 y^{^{'}} + 7 t = t^{3} e^{4 t} cos (t)

\frac{d ^{2} y}{d x ^{2}} - y = sin^{2} (x)

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