What is the solution for y^{''}+6y^'+9y=t^{-2}e^{-3t} ?

The solution for y^{''}+6y^'+9y=t^{-2}e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}-e^{-3t}ln(t)-e^{-3t}

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y^{''}+6y^'+9y=t^{-2}e^{-3t}

y^{^{''}} + 6 y^{^{'}} + 9 y = t^{- 2} e^{- 3 t}

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} - e^{- 3 t} ln (t) - e^{- 3 t}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 9 y = t^{- 2} e^{- 3 t}

Solve linear ODE:

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} - e^{- 3 t} ln (t) - e^{- 3 t}

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} - e^{- 3 t} ln (t) - e^{- 3 t}

y^{^{'}} = \frac{3}{( x + 2 ) ^{2}}

7^{x + a r c c o s (x)}

\int_{\frac{π}{5}}^{\infty} \frac{5}{x} d x

f^{^{'}} (x) = \frac{3}{x}

\int \frac{1}{1 + ln ( x )} d x

What is the solution for y^{''}+6y^'+9y=t^{-2}e^{-3t} ?
The solution for y^{''}+6y^'+9y=t^{-2}e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}-e^{-3t}ln(t)-e^{-3t}