MHB How to Solve This ODE with Substitution?

[jasonmcc]

I haven't done ODEs in a while nor have a book handing.

How do I tackle an equation of the form
\[
2xyy'=-x^2-y^2
\]
I tried polar but that didn't seem to work.

 

[chisigma]

Re: separable or not separable ODE

I haven't done ODEs in a while nor have a book handing.

How do I tackle an equation of the form
\[
2xyy'=-x^2-y^2
\]
I tried polar but that didn't seem to work.

With simple steps Yoy arrive to write...

$\displaystyle (x^{2} + y^{2})\ dx + 2\ x\ y\ d y =0\ (1)$ ... and the expression (1) is an 'exact differential'... Kind regards $\chi$ $\sigma$

 

[MarkFL]

Re: separable or not separable ODE

Another approach would be to write the ODE as:

$$\frac{dy}{dx}=-\frac{1}{2}\left(\frac{x}{y}+\frac{y}{x} \right)$$

and use the substitution:

$$v=\frac{y}{x}$$