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separable or non-separable ODE

wmccunes

New member
Jan 30, 2013
8
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

Well-known member
Feb 13, 2012
1,704
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

Administrator
Staff member
Feb 24, 2012
13,775
Re: separable or not separable ODE

Another approach would be to write the ODE as:

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

and use the substitution:

\(\displaystyle v=\frac{y}{x}\)