(adsbygoogle = window.adsbygoogle || []).push({}); Differential Equation w/ Homogeneous Coefficients - y=ux substitution

I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class.

1. The problem statement, all variables and given/known data

(x+y){dx} - (x-y){dy} = 0

2. Relevant equations

y=ux, {dy} = u{dx} + x{du}

3. The attempt at a solution

Substitution should lead to a separable equation in x and u:

(x+ux){dx} + (ux - x)(u{dx} + x{du}) = 0;

x(u+1)dx + u^{2}x{dx} + ux^{2}{du} - ux{dx} - x^{2}{du} = 0;

xu(^{2}){dx} + x^{2}(u - 1){du} = 0;

-1/x{dx} = (u - 1)/(u^{2}+ 1){du}

Okay, I am assuming that I am correct up to this point but the answer given by the text is:

Arc tan(y/x) - 1/2log(x^{2}+ y^{2}) = c

I understand where the Arc tan(y/x) comes from - the (1/(u^{2}+ 1)). I am having trouble with the 1/2 log(x^{2}+ y^{2}) - where does the -log(x) go?

I have a couple of other problems in the same form as this which arise in isogonal trajectories and I don't want to just skip over this because I am obviously having a problem with these integrations.

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# Equation w/ Homogeneous Coefficients - y=ux substitution

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