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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} + 4 y^{^{'}} = 0, y (0) = 1, y^{^{'}} (0) = 1, y^{^{''}} (0) = 1

144 y^{^{''}} - 24 y^{^{'}} + y = 0

y^{^{''}} - 5 y^{^{'}} + 6 y = 0, y (0) = - \frac{8}{5}, y^{^{'}} (0) = - 5

y^{^{''}} + 4 y^{^{'}} + 5 y = 0, y (0) = 2, y (0) = - 7

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

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