What is the solution for 4(d^2x)/(dt^2)+12(dx)/(dt)+9x=0 ?

The solution for 4(d^2x)/(dt^2)+12(dx)/(dt)+9x=0 is x=c_{1}e^{-(3t)/2}+c_{2}e^{-(3t)/2}t

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4(d^2x)/(dt^2)+12(dx)/(dt)+9x=0

4 \frac{d ^{2} x}{d t ^{2}} + 12 \frac{d x}{d t} + 9 x = 0

x = c_{1} e^{- \frac{3 t}{2}} + c_{2} e^{- \frac{3 t}{2}} t

Solution steps

4 \frac{d ^{2} x}{d t ^{2}} + 12 \frac{d x}{d t} + 9 x = 0

Solve linear ODE:

x = c_{1} e^{- \frac{3 t}{2}} + c_{2} e^{- \frac{3 t}{2}} t

x = c_{1} e^{- \frac{3 t}{2}} + c_{2} e^{- \frac{3 t}{2}} t

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

\frac{d ^{2} y}{d x ^{2}} - 25 y = 0

9 y^{^{''}} + 6 y^{^{'}} + 82 y = 0, y (0) = 2

y^{^{''}} - 9 y = 0, y (0) = 2, y^{^{'}} (0) = 4

(4 D^{2} - 3 D - 10) y = 0

What is the solution for 4(d^2x)/(dt^2)+12(dx)/(dt)+9x=0 ?
The solution for 4(d^2x)/(dt^2)+12(dx)/(dt)+9x=0 is x=c_{1}e^{-(3t)/2}+c_{2}e^{-(3t)/2}t