What is the solution for y^{''}+8y^'+16y=0,y(0)=1,y^'(0)=0 ?

The solution for y^{''}+8y^'+16y=0,y(0)=1,y^'(0)=0 is y=e^{-4t}+4e^{-4t}t

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y^{''}+8y^'+16y=0,y(0)=1,y^'(0)=0

y^{^{''}} + 8 y^{^{'}} + 16 y = 0, y (0) = 1, y^{^{'}} (0) = 0

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + y^{^{'}} + 3 y = 0

y^{^{''}} + 225 y = 0

y^{^{''}} + 4 y^{^{'}} + 5 y = 0, y (0) = 2, y^{^{'}} (0) = - 7

2 y^{^{''}} + 2 y^{^{'}} + 3 y = 0

What is the solution for y^{''}+8y^'+16y=0,y(0)=1,y^'(0)=0 ?
The solution for y^{''}+8y^'+16y=0,y(0)=1,y^'(0)=0 is y=e^{-4t}+4e^{-4t}t