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

[tex]x^{2}\frac{dy}{dx}-2xy=3y^{4}[/tex]

2. Relevant equations

Bernoulli's equation

3. The attempt at a solution

[tex]x^{2}\frac{dy}{dx}-2xy=3y^{4}[/tex]

First I divide through by [tex]x^{2}[/tex] and [tex]y^{n}[/tex] to put it in standard form, and then to begin Bernoulli's equation process.

[tex]y^{-4}y'-\frac{2}{x}y^{-3}=\frac{3}{x^{2}}[/tex]

u-substitution:

[tex]w=y^{1-n}=y^{-3}[/tex]

[tex]w'=-3y^{-4}y'[/tex]

Substitute in...

[tex]\frac{-1}{3}w'-\frac{2}{x}w=\frac{3}{x^{2}}[/tex]

Put in Standard Form...

[tex]w'+\frac{6}{x}w=-\frac{9}{x^{2}}[/tex]

Get the integrating factor...

[tex]p(x)=\frac{6}{x} \Rightarrow \int\frac{6}{x}dx = 6ln(x)\Rightarrow\mu=e^{6lnx}=x^{6}[/tex]

[tex]\int\frac{d}{dx}(w*x^{6})dx=\int\frac{-9}{x^{4}}dx[/tex]

[tex]w=\frac{3}{x^{9}}+cx^{-6}[/tex]

I'll stop here since I don't think the simplification with help much (w->y^-3)

Wolfram got this:

Clickity

Which kinda-sorta looks like my answer, except the powers are off. Where did I go wrong?

Using wolfram to simplify, with y^-3 plugged in for w

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