What is the solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-5 ?

The solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-5 is y=2e^{-3t}+e^{-3t}t

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y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-5

y^{^{''}} + 6 y^{^{'}} + 9 y = 0, y (0) = 2, y^{^{'}} (0) = - 5

y = 2 e^{- 3 t} + e^{- 3 t} t

Solution steps

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

Solve linear ODE:

y = 2 e^{- 3 t} + e^{- 3 t} t

y = 2 e^{- 3 t} + e^{- 3 t} t

(d^{2} - 2 d + 5)^{2} y = 0

3 x y^{^{'}} = 3 x y^{^{''}}

y^{^{''}} - (2 α - 12) y^{^{'}} + (α^{2} - 12 α + 32) y = 0

3 y^{^{''}} + 12 y = 0, y (0) = 5, y^{^{'}} (0) = 12

y = \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x}

What is the solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-5 ?
The solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-5 is y=2e^{-3t}+e^{-3t}t