Non-linear ODE with pursuit curves

[abercrombiems02]

How would one go about finding an analytical solution to the following ODE

note that we are trying to find y(x) subject to...

(x*y'')^2 - (1 + (y')^2) = 0

 

[HallsofIvy]

How would one go about finding an analytical solution to the following ODE

note that we are trying to find y(x) subject to...

(x*y'')^2 - (1 + (y')^2) = 0

The first thing I notice is that there is no "y" in that equation- just
y" and y'. If you let u= y' then you have the pair of first order equations
(x*u')^2- (1+ u^2)= 0 or
$$xu'= \pm\sqrt{1+ u^2}$$
That gives two relatively simple separable equations.