What is the solution for y^{''}+8y^'+16y=0,y(0)=5,y^'(0)=0 ?

The solution for y^{''}+8y^'+16y=0,y(0)=5,y^'(0)=0 is y=5e^{-4t}+20te^{-4t}

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y^{''}+8y^'+16y=0,y(0)=5,y^'(0)=0

y^{^{''}} + 8 y^{^{'}} + 16 y = 0, y (0) = 5, y^{^{'}} (0) = 0

y = 5 e^{- 4 t} + 20 t e^{- 4 t}

Solution steps

y^{''} (t) + 8 y^{'} (t) + 16 y = 0

Solve linear ODE:

y = 5 e^{- 4 t} + 20 t e^{- 4 t}

y = 5 e^{- 4 t} + 20 t e^{- 4 t}

y^{^{''}} + y^{^{'}} - 20 y = 0, y (0) = 9, y (1) = 2

9 y^{^{''}} + 5 y^{^{'}} - 4 y = 0

(y^{^{''''}} + 8 y^{^{''}} + 16 y)^{2} = 0

y^{^{''}} + 5 y^{^{'}} + 4 y = 0, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} = 64 y

What is the solution for y^{''}+8y^'+16y=0,y(0)=5,y^'(0)=0 ?
The solution for y^{''}+8y^'+16y=0,y(0)=5,y^'(0)=0 is y=5e^{-4t}+20te^{-4t}