What is the solution for y^{''}+12y^'+27y=0 ?

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

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y^{''}+12y^'+27y=0

y^{^{''}} + 12 y^{^{'}} + 27 y = 0

y = c_{1} e^{- 3 t} + c_{2} e^{- 9 t}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- 3 t} + c_{2} e^{- 9 t}

y = c_{1} e^{- 3 t} + c_{2} e^{- 9 t}

y^{^{'''}} - 6 y^{^{''}} - 15 y^{^{'}} + 100 y = 0

4 y^{^{''}} - 3 y^{^{'}} - 7 y = 0

y^{^{''}} + 4 y = 0, y (\frac{π}{6}) = 4, y^{^{'}} (\frac{π}{6}) = - 8

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

y^{^{'''''}} + 2 y^{^{''''}} + 4 y^{^{'''}} + 8 y^{^{''}} = 0

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