What is the solution for y^{''}+2*y^'+y=e^{-t} ?

The solution for y^{''}+2*y^'+y=e^{-t} is y=c_{1}e^{-t}+c_{2}te^{-t}+(e^{-t}t^2)/2

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y^{''}+2*y^'+y=e^{-t}

y^{^{''}} + 2 \cdot y^{^{'}} + y = e^{- t}

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

Solution steps

y^{''} (t) + 2 \cdot y^{'} (t) + y = e^{- t}

Solve linear ODE:

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

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

n = 0 \sum \infty \frac{1}{2 ^{n} n !}

f (x) = \frac{( x ^{3} + 1 )}{( x ^{4} + 1 )}

x \to - 1 lim (\frac{x + 1}{x ^{2} - 1})

\int 2 4 - x^{2} d x

x \to π lim (\frac{( e ^{s i n (x)} - 1 )}{x - π})

What is the solution for y^{''}+2*y^'+y=e^{-t} ?
The solution for y^{''}+2*y^'+y=e^{-t} is y=c_{1}e^{-t}+c_{2}te^{-t}+(e^{-t}t^2)/2