What is the solution for y^{''}+12y^'+27y=0,y(0)=12,y^'(0)=-84 ?

The solution for y^{''}+12y^'+27y=0,y(0)=12,y^'(0)=-84 is y=4e^{-3t}+8e^{-9t}

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y^{''}+12y^'+27y=0,y(0)=12,y^'(0)=-84

y^{^{''}} + 12 y^{^{'}} + 27 y = 0, y (0) = 12, y^{^{'}} (0) = - 84

y = 4 e^{- 3 t} + 8 e^{- 9 t}

Solution steps

y^{''} (t) + 12 y^{'} (t) + 27 y = 0

Solve linear ODE:

y = 4 e^{- 3 t} + 8 e^{- 9 t}

y = 4 e^{- 3 t} + 8 e^{- 9 t}

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

y^{^{''}} + 10 y^{^{'}} = - 21 y

2 (D^{4} + 11 D^{3} - 4 D^{2} - 69 D + 34) y = 0

y^{^{''}} + 2 y + 3 y = 0

4 y^{^{''}} + 32 y^{^{'}} + 113 y = 0

What is the solution for y^{''}+12y^'+27y=0,y(0)=12,y^'(0)=-84 ?
The solution for y^{''}+12y^'+27y=0,y(0)=12,y^'(0)=-84 is y=4e^{-3t}+8e^{-9t}