What is the solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-1 ?

The solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-1 is y=2e^{-3t}+5te^{-3t}

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y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-1

y^{^{''}} + 6 y^{^{'}} + 9 y = 0, y (0) = 2, y^{^{'}} (0) = - 1

y = 2 e^{- 3 t} + 5 t e^{- 3 t}

Solution steps

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

Solve linear ODE:

y = 2 e^{- 3 t} + 5 t e^{- 3 t}

y = 2 e^{- 3 t} + 5 t e^{- 3 t}

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

(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

What is the solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-1 ?
The solution for y^{''}+6y^'+9y=0,y(0)=2,y^'(0)=-1 is y=2e^{-3t}+5te^{-3t}