What is the solution for y^{''}+6y^'+9y=0,y(0)=9,y^'(0)=2 ?

The solution for y^{''}+6y^'+9y=0,y(0)=9,y^'(0)=2 is y=9e^{-3t}+29te^{-3t}

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y^{''}+6y^'+9y=0,y(0)=9,y^'(0)=2

y^{^{''}} + 6 y^{^{'}} + 9 y = 0, y (0) = 9, y^{^{'}} (0) = 2

y = 9 e^{- 3 t} + 29 t e^{- 3 t}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 9 y = 0

Solve linear ODE:

y = 9 e^{- 3 t} + 29 t e^{- 3 t}

y = 9 e^{- 3 t} + 29 t e^{- 3 t}

y^{^{''}} + 3 y^{^{'}} - 9 y = 0

100 y^{^{''}} + 90 y^{^{'}} - 162 y = 0

y^{^{''}} - 4 y^{^{'}} - 13 y = 0

\frac{3}{4} \frac{d ^{2} x}{d t ^{2}} + 72 x = 0, x (0) = - \frac{1}{2}

y^{^{''}} - 16 y^{^{'}} + 73 y = 0

What is the solution for y^{''}+6y^'+9y=0,y(0)=9,y^'(0)=2 ?
The solution for y^{''}+6y^'+9y=0,y(0)=9,y^'(0)=2 is y=9e^{-3t}+29te^{-3t}