What is the solution for (d^2y)/(dx^2)+2(dy)/(dx)+y=e^{-x} ?

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

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(d^2y)/(dx^2)+2(dy)/(dx)+y=e^{-x}

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

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

Solution steps

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

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

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

\frac{d ^{2} x}{d t ^{2}} - 6 \frac{d x}{d t} - 7 x = 6

\frac{1}{6} y^{^{''}} + 6 y = 2 tan (6 t) - \frac{1}{3} e^{t}

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