(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If y(1+x^{2}) dy/dx = 2x (1-y^{2}), prove that (1+x^{2})^{2}(1-y^{2})=A, where A is constant.

2. Relevant equations

Separable equations

3. The attempt at a solution

Separate the terms:

y/(1-y^{2}) dy = 2x/(1+x^{2}) dx

Integrating both sides will get:

∫ y/(1-y^{2}) dy = ∫ 2x/(1+x^{2}) dx

Use substitution method for ∫ y/(1-y^{2}) dy:

u = 1-y^{2}

du = -2y dy

-du/2 = y dy

∫ -u/2 du = -1/2 ∫ u du

= (-1/2)*(u^{2}/2)

= -u^{2}/4 + C

= -(1-y^{2})^{2}/4

Use substitution method for ∫ 2x/(1+x^{2}) dx:

u= 1+x^{2}

du = 2x

∫ 1/u du = ln u + C

= ln (1+x^{2})

Putting them back together will get:

-(1-y^{2})^{2}/4 = ln (1+x^{2})

I'm pretty much unable to continue from here.

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# Separable equations proving

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