What is the solution for (d^2y)/(dt^2)+8(dy)/(dt)+16y=0,y(0)=5 ?

The solution for (d^2y)/(dt^2)+8(dy)/(dt)+16y=0,y(0)=5 is y=5e^{-4t}+c_{2}te^{-4t}

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(d^2y)/(dt^2)+8(dy)/(dt)+16y=0,y(0)=5

\frac{d ^{2} y}{d t ^{2}} + 8 \frac{d y}{d t} + 16 y = 0, y (0) = 5

y = 5 e^{- 4 t} + c_{2} t e^{- 4 t}

Solution steps

\frac{d ^{2} y}{d t ^{2}} + 8 \frac{d y}{d t} + 16 y = 0

Solve linear ODE:

y = 5 e^{- 4 t} + c_{2} t e^{- 4 t}

y = 5 e^{- 4 t} + c_{2} t e^{- 4 t}

y^{^{''''}} + 4 y^{^{'''}} + 13 y^{^{''}} + 36 y^{^{'}} + 36 y = 0

y^{^{''}} + 9 y = 0, y (0) = 3, y^{^{'}} (0) = - 9

\frac{d ^{2} x}{d t ^{2}} + 4 \cdot \frac{d x}{d t} + 8 x = 0

(4 D^{3} - 5 D) y = 0

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

What is the solution for (d^2y)/(dt^2)+8(dy)/(dt)+16y=0,y(0)=5 ?
The solution for (d^2y)/(dt^2)+8(dy)/(dt)+16y=0,y(0)=5 is y=5e^{-4t}+c_{2}te^{-4t}