Solve the given differential eq by using an appropriate sub

[hank]

Ok, here's the problem...

dy/dx = (y - x)/(y + x)
(y + x) dy = (y - x)dx //separating variables

I'll choose y = ux for the sub, since neither seem simpler.

y = ux => dy = xdu + udx //By differentiation and using product rule

(ux + x)(xdu + udx) = (ux - x) dx // By substitution

(ux^2 du + xu^2 dx + x^2 du + ux dx = ux dx - x dx // Factoring

(u + 1)x^2 du = -x(1 + u^2) dx // Collecting the like terms and simplifying

(u + 1)/(1 + u^2) du = -1/x dx // separating variables again

And this is the point where I invariably get stuck trying to integrate both sides.

S (u + 1)/(1 + u^2) du = -ln|x| + C //integrate the right side


Can someone verify I'm ok up to this point and then show me how to integrate the left side? I attempt to integrate by parts, but I keep making a mistake somewhere that gets me in trouble.

Best Regards,
--Hank Stalica

 

[royhaas]

For the Left Hand Side, break it up into two integrals.

 

[Deleted member 4993]

S (u + 1)/(1 + u^2) du

\(\displaystyle \int \frac{u+1}{1+u^2} du\)

\(\displaystyle = \, \int \frac{u}{1+u^2} du \, + \, \int \frac{1}{1+u^2} du\)

\(\displaystyle = \, \frac{1}{2}\cdot ln(1+u^2) \, + \, \tan^{-1}(u) \, + \, C\)

 

[hank]

Thanks for the replies, everyone.

I got it!

I guess I just had a mental block on breaking the integral into two pieces.
Once I split it up, it all fell into place.