What is the solution for y^{'''}-3y^{''}+9y^'+13y=0 ?

The solution for y^{'''}-3y^{''}+9y^'+13y=0 is y=c_{1}e^{-t}+e^{2t}(c_{2}cos(3t)+c_{3}sin(3t))

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y^{'''}-3y^{''}+9y^'+13y=0

y^{^{'''}} - 3 y^{^{''}} + 9 y^{^{'}} + 13 y = 0

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

Solution steps

y^{'''} (t) - 3 y^{''} (t) + 9 y^{'} (t) + 13 y = 0

Solve linear ODE:

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

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

y^{^{''}} + 2 y^{^{'}} + 2 y = 0, y (\frac{π}{2}) = 0, y^{^{'}} (\frac{π}{2}) = 3

2 y^{^{''}} - 2 y^{^{'}} - y = 0

y^{^{''}} - 3 y^{^{'}} + (2 + a) y = 0

y^{^{''}} - 11 y^{^{'}} = 0

9 y^{^{''}} + 4 y^{^{'}} = 0

What is the solution for y^{'''}-3y^{''}+9y^'+13y=0 ?
The solution for y^{'''}-3y^{''}+9y^'+13y=0 is y=c_{1}e^{-t}+e^{2t}(c_{2}cos(3t)+c_{3}sin(3t))