What is the solution for y^{'''}+12y^{''}+45y^'=0 ?

The solution for y^{'''}+12y^{''}+45y^'=0 is y=c_{1}+e^{-6t}(c_{2}cos(3t)+c_{3}sin(3t))

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y^{'''}+12y^{''}+45y^'=0

y^{^{'''}} + 12 y^{^{''}} + 45 y^{^{'}} = 0

y = c_{1} + e^{- 6 t} (c_{2} cos (3 t) + c_{3} sin (3 t))

Solution steps

y^{'''} (t) + 12 y^{''} (t) + 45 y^{'} (t) = 0

Solve linear ODE:

y = c_{1} + e^{- 6 t} (c_{2} cos (3 t) + c_{3} sin (3 t))

y = c_{1} + e^{- 6 t} (c_{2} cos (3 t) + c_{3} sin (3 t))

y^{^{''}} - 2 y^{^{'}} - 3 y = 0, y (0) = 3, y^{^{'}} (0) = 13

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

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

What is the solution for y^{'''}+12y^{''}+45y^'=0 ?
The solution for y^{'''}+12y^{''}+45y^'=0 is y=c_{1}+e^{-6t}(c_{2}cos(3t)+c_{3}sin(3t))