What is the solution for y^{''}+6y^'+5y=t*e^{-t} ?

The solution for y^{''}+6y^'+5y=t*e^{-t} is y=c_{1}e^{-t}+c_{2}e^{-5t}+(e^{-t}t^2)/8-(e^{-t}t)/(16)

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y^{''}+6y^'+5y=t*e^{-t}

y^{^{''}} + 6 y^{^{'}} + 5 y = t \cdot e^{- t}

y = c_{1} e^{- t} + c_{2} e^{- 5 t} + \frac{e ^{- t} t ^{2}}{8} - \frac{e ^{- t} t}{16}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 5 y = t \cdot e^{- t}

Solve linear ODE:

y = c_{1} e^{- t} + c_{2} e^{- 5 t} + \frac{e ^{- t} t ^{2}}{8} - \frac{e ^{- t} t}{16}

y = c_{1} e^{- t} + c_{2} e^{- 5 t} + \frac{e ^{- t} t ^{2}}{8} - \frac{e ^{- t} t}{16}

y^{^{''}} - 6 y^{^{'}} - 16 y = - 24 t + 32 t^{2}

16 y^{^{''}} + y = 4 sec (\frac{π}{4})

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

y^{^{''}} + y^{^{'}} - 8 y = 48 t

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

What is the solution for y^{''}+6y^'+5y=t*e^{-t} ?
The solution for y^{''}+6y^'+5y=t*e^{-t} is y=c_{1}e^{-t}+c_{2}e^{-5t}+(e^{-t}t^2)/8-(e^{-t}t)/(16)