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Homework Statement
[tex]x^{2}\frac{dy}{dx}-2xy=3y^{4}[/tex]
Homework Equations
Bernoulli's equation
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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