What is the solution for (d^2x)/(dt^2)+3(dx)/(dt)+8x=0,x=0,t=0 ?

The solution for (d^2x)/(dt^2)+3(dx)/(dt)+8x=0,x=0,t=0 is x=c_{2}e^{-(3t)/2}sin((sqrt(23)t)/2)

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(d^2x)/(dt^2)+3(dx)/(dt)+8x=0,x=0,t=0

\frac{d ^{2} x}{d t ^{2}} + 3 \frac{d x}{d t} + 8 x = 0, x = 0, t = 0

x = c_{2} e^{- \frac{3 t}{2}} sin (\frac{23 t}{2})

Solution steps

\frac{d ^{2} x}{d t ^{2}} + 3 \frac{d x}{d t} + 8 x = 0

Solve linear ODE:

x = c_{2} e^{- \frac{3 t}{2}} sin (\frac{23 t}{2})

x = c_{2} e^{- \frac{3 t}{2}} sin (\frac{23 t}{2})

y^{^{''}} + a \cdot y = 0

y^{^{''}} - 4 c^{2} y = 0

\frac{d ^{2} y}{d t ^{2}} + 5 \frac{d y}{d t} + 4 y = 0

y^{^{'''}} - 2 y^{^{''}} - 11 y^{^{'}} + 12 y = 0

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

What is the solution for (d^2x)/(dt^2)+3(dx)/(dt)+8x=0,x=0,t=0 ?
The solution for (d^2x)/(dt^2)+3(dx)/(dt)+8x=0,x=0,t=0 is x=c_{2}e^{-(3t)/2}sin((sqrt(23)t)/2)