What is the solution for y^{'''}+5y^{''}+25y^'+125y=0 ?

The solution for y^{'''}+5y^{''}+25y^'+125y=0 is y=c_{1}e^{-5t}+c_{2}cos(5t)+c_{3}sin(5t)

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y^{'''}+5y^{''}+25y^'+125y=0

y^{^{'''}} + 5 y^{^{''}} + 25 y^{^{'}} + 125 y = 0

y = c_{1} e^{- 5 t} + c_{2} cos (5 t) + c_{3} sin (5 t)

Solution steps

y^{'''} (t) + 5 y^{''} (t) + 25 y^{'} (t) + 125 y = 0

Solve linear ODE:

y = c_{1} e^{- 5 t} + c_{2} cos (5 t) + c_{3} sin (5 t)

y = c_{1} e^{- 5 t} + c_{2} cos (5 t) + c_{3} sin (5 t)

\frac{d ^{2} y}{d t ^{2}} + 14 \frac{d y}{d t} + 48 y = 0

y^{^{''}} + 2 y^{^{''}} + 10 y = 0, y (0) = - 2, y^{^{'}} (0) = 0

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

5 f^{^{''}} (t) + 50 f^{^{'}} (t) + f (t) = 0

\frac{d ^{2} y}{d x ^{2}} = - 6 y

What is the solution for y^{'''}+5y^{''}+25y^'+125y=0 ?
The solution for y^{'''}+5y^{''}+25y^'+125y=0 is y=c_{1}e^{-5t}+c_{2}cos(5t)+c_{3}sin(5t)