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

I must calculate the general solution to the following (reducible to homogeneous) DE:

[itex](x^2y^2-1)y'+2xy^3=0[/itex].

Hint: Use the substitution [itex]y=z^\alpha[/itex].

2. Relevant equations

The one given in the hint.

3. The attempt at a solution

So I've used the hint. This gave me [itex](x^2z^{2 \alpha }-1)\alpha z^{\alpha -1}+2xz^{3\alpha }=0[/itex].

I factorized by [itex]z^{\alpha -1}[/itex] and I reach that [itex]z^{\alpha -1}=0[/itex] (thus z=0, thus y=0) or [itex]z^{2\alpha } (\alpha x^2 +2xz)=\alpha[/itex].

I ran out of ideas on the second condition. ( I don't think back-substituting [itex]z^\alpha =y[/itex] will help).

Any help is appreciated. Thanks!

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# Solving a DE

Can you offer guidance or do you also need help?

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