What is the solution for y^{''}+12y^'+37y=0 ?

The solution for y^{''}+12y^'+37y=0 is y=e^{-6t}(c_{1}cos(t)+c_{2}sin(t))

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y^{''}+12y^'+37y=0

y^{^{''}} + 12 y^{^{'}} + 37 y = 0

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

Solution steps

y^{''} (t) + 12 y^{'} (t) + 37 y = 0

Solve linear ODE:

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

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

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

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

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

(D^{2} - 2 D + 2) y = 0

4 \frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} = - y

What is the solution for y^{''}+12y^'+37y=0 ?
The solution for y^{''}+12y^'+37y=0 is y=e^{-6t}(c_{1}cos(t)+c_{2}sin(t))