What is the solution for (d^2x)/(dt^2)+6(dx)/(dt)+9x=0 ?

The solution for (d^2x)/(dt^2)+6(dx)/(dt)+9x=0 is x=c_{1}e^{-3t}+c_{2}te^{-3t}

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(d^2x)/(dt^2)+6(dx)/(dt)+9x=0

\frac{d ^{2} x}{d t ^{2}} + 6 \frac{d x}{d t} + 9 x = 0

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

Solution steps

\frac{d ^{2} x}{d t ^{2}} + 6 \frac{d x}{d t} + 9 x = 0

Solve linear ODE:

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

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

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

y^{^{''''}} + y^{^{''}} + 9 y = 0

y^{^{''}} - 7 y^{^{'}} + 10 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

4 x^{^{''}} + 2 x = 0

y^{^{''}} - 10 y^{^{'}} - 98 y = 0

What is the solution for (d^2x)/(dt^2)+6(dx)/(dt)+9x=0 ?
The solution for (d^2x)/(dt^2)+6(dx)/(dt)+9x=0 is x=c_{1}e^{-3t}+c_{2}te^{-3t}