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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

\frac{d}{d x} (36 x^{2} - 72 x)

x \to 0 lim (cos (\frac{π}{x}))

\int \frac{ln ( x )}{x ^{14}} d x

\int (1 + sin (θ))^{2} d θ

x \to - 2 + lim (x^{2} - 5)

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