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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 8 y^{^{'}} + 32 y = 0, y^{0} = 1, y^{^{'}} 0 = - 4

y^{^{''}} + 7 y^{^{'}} - 12 y = 0

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

y^{^{''}} + y = 0, y (0) = 3, y (0) = 30

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

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