# How to solve the non-linear second order ODE $y''=\frac{x}{y}-1$

How can I find a general solution to the non-linear second order ODE $$y'' = \frac{x}{y}-1,$$ if there is one expressible in closed form?

So far I have only found the particular solution $y(x)=x$.

• We need to know what you have tried and where you get stuck. Jan 18, 2017 at 18:05
• Was there anything more specific in the question? Like the general solution is in the form... Jan 18, 2017 at 18:12
• @TheCount Well, in class we mentioned some 6 types of hgher order nonlinear ODEs which we can solve, but I can't fit this one into any of those types, as they all assume that we have y' as well as y''. Jan 18, 2017 at 18:19
• @TehRod There was nothing more specific. It also wasn't stated that a general solution can be found. Jan 18, 2017 at 18:21
• This question looks like a pretty hard problem that very unlikely has an explicit solution. Jan 18, 2017 at 18:24

• This seems to give a possible solution, however it's full of typo-like mistakes. I'm having problems understanding how he got the z(1-xz) term in (3.4). I always get just z(1-z) without the x. As far as i can tell, later on he uses that result so it shouldn't be a typo? Jan 19, 2017 at 14:14

HINT

Can we try introducing $u(x,y)$ differentiating along these lines

$$y^{\prime2} = x^2 -u(x,y)$$

$$2 y{\prime} y{\prime \prime} = 2 x - \left( \frac{ \partial u}{ \partial x} + \frac{\partial u} {\partial y} \frac{dy} {dx} \right)$$

So may be the pde that needs to be solved is

$$2y =\frac{\partial u} {\partial x} + \frac{\partial u} {\partial y} \frac{dy} {dx}$$

• Unfortunately, I'm taking a course in ordinary DEs, where this problem was asked, so I am not familiar with PDEs and I am not sure what you are getting at. Can you explain a bit more? Jan 18, 2017 at 19:09