What is the solution for y^{''}+12y^'+27y=0,y(0)=1,y^'(0)=-12 ?

The solution for y^{''}+12y^'+27y=0,y(0)=1,y^'(0)=-12 is y=(1-3/2)e^{-3t}+3/2 e^{-9t}

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y^{''}+12y^'+27y=0,y(0)=1,y^'(0)=-12

y^{^{''}} + 12 y^{^{'}} + 27 y = 0, y (0) = 1, y^{^{'}} (0) = - 12

y = (1 - \frac{3}{2}) e^{- 3 t} + \frac{3}{2} e^{- 9 t}

Solution steps

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

Solve linear ODE:

y = (1 - \frac{3}{2}) e^{- 3 t} + \frac{3}{2} e^{- 9 t}

y = (1 - \frac{3}{2}) e^{- 3 t} + \frac{3}{2} e^{- 9 t}

18 y^{^{''}} + 9 y^{^{'}} - 5 y = 0

y^{^{''}} + 2 y^{^{'}} + 5 y = 0, y (0) = 3, y^{^{'}} (0) = 4

y^{^{'''}} + y^{^{''}} - 4 y^{^{'}} + 6 y = 0

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

\frac{d ^{2} y}{d x ^{2}} (1) - 9 y = 0

What is the solution for y^{''}+12y^'+27y=0,y(0)=1,y^'(0)=-12 ?
The solution for y^{''}+12y^'+27y=0,y(0)=1,y^'(0)=-12 is y=(1-3/2)e^{-3t}+3/2 e^{-9t}