What is the solution for y^{''}+16y^'+64y=0,y(0)=1,y^'(0)=0 ?

The solution for y^{''}+16y^'+64y=0,y(0)=1,y^'(0)=0 is y=e^{-8t}+8e^{-8t}t

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y^{''}+16y^'+64y=0,y(0)=1,y^'(0)=0

y^{^{''}} + 16 y^{^{'}} + 64 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y = e^{- 8 t} + 8 e^{- 8 t} t

Solution steps

y^{''} (t) + 16 y^{'} (t) + 64 y = 0

Solve linear ODE:

y = e^{- 8 t} + 8 e^{- 8 t} t

y = e^{- 8 t} + 8 e^{- 8 t} t

y^{^{''}} - 4 y^{^{'}} = 0, y (0) = 6 - 6 e^{4}, y (1) = 0

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

y^{^{''}} - \frac{3}{2} y^{^{'}} - y = 0

y^{^{''}} - 8 y^{^{'}} = 0, y (0) = 5 - 5 e^{8}, y (1) = 0

y^{^{''}} - 8 y^{^{'}} + 10 y = 0, y (0) = 0, y^{^{'}} (0) = 7

What is the solution for y^{''}+16y^'+64y=0,y(0)=1,y^'(0)=0 ?
The solution for y^{''}+16y^'+64y=0,y(0)=1,y^'(0)=0 is y=e^{-8t}+8e^{-8t}t