# A second order nonlinear ordinary differential equation

How, if possible, could I solve the equation: x''x=((x')^2)/2? Thanks.

Filip Larsen
Gold Member
If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.

Try to find such a pair of real numbers a & b that d/dt (x^a x'^b) is proportional to x"x-((x')^2)/2.

If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.

I've tried that but is there a way to actually solve it without guessing?

phyzguy
Equations like this, which do not contain the independent variable, are called autonomous systems. They are easily solved with the substitution x'' = x' dx'/dx . Then collect the terms in x' and x and integrate them both. This will give you x' in terms of x, which you can then integrate a second time.

Do you only require a numerical solution given some particular boundry conditions?

rearrange:

$$\frac{1}{2x}=\frac{\ddot{x}}{x}=-\frac{d}{dx}(\frac{1}{\dot{x}})$$

$$dt=(C-ln(\sqrt{x}))dx =d[x(C'-ln(\sqrt{x}))]$$