What is the solution for y^{''}+6y^'+10y=0,y(0)=0,y^'(0)=6 ?

The solution for y^{''}+6y^'+10y=0,y(0)=0,y^'(0)=6 is y=6e^{-3t}sin(t)

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y^{''}+6y^'+10y=0,y(0)=0,y^'(0)=6

y^{^{''}} + 6 y^{^{'}} + 10 y = 0, y (0) = 0, y^{^{'}} (0) = 6

y = 6 e^{- 3 t} sin (t)

Solution steps

y^{''} (t) + 6 y^{'} (t) + 10 y = 0

Solve linear ODE:

y = 6 e^{- 3 t} sin (t)

y = 6 e^{- 3 t} sin (t)

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

49 y^{^{''}} - 49 y = 0

y^{^{''}} + 16 y = 0, y^{^{'}} (0) = 1

\frac{d ^{2} y}{d x ^{2}} - 4 (\frac{d y}{d x}) - 32 y = 0

y^{^{''}} + 8 y^{^{'}} + 20 y = 0, y (0) = 1

What is the solution for y^{''}+6y^'+10y=0,y(0)=0,y^'(0)=6 ?
The solution for y^{''}+6y^'+10y=0,y(0)=0,y^'(0)=6 is y=6e^{-3t}sin(t)