<s>[INST] Solving a differential equation with Bernoulli's Method What approach do you take to solve the differential equation $ y' + (6y/x) = (y^3)/ x^5\ $ through the use of Bernoulli's method?
I've assumed u = y^(-2) for substitution, but I don't know where to go from there. The answer is  y $ ((C\x^2\) + (1/8x^4))^(-.5) \ $
 [/INST] $\frac{dy}{dx}+6\frac{y}{x}=\frac{y^3}{x^5}$
Multiply by $\frac{1}{y^3}$
$\frac{1}{y^3}\frac{dy}{dx}+6\frac{1}{y^2x}=\frac{1}{x^5}$
Let $\frac{1}{y^2}=z$
$\frac{-2}{y^3}\frac{dy}{dx}=\frac{dz}{dx}$
$-\frac{dz}{dx}-12\frac{z}{x}=-2\frac{1}{x^5}$
$P(x)=-\frac{12}{x}$
$\int e^{-\frac{12}{x}}dx=x^{-12}$
${z}{x^{-12}}=\int \frac{1}{x^{12}}\frac{1}{x^5}dx$
${z}{x^{-12}}=-\frac{1}{16 x^{16}}+C$
$z=-\frac{1}{16x^4}+Cx^{12}$
I hope tou can continue by replacing $z=\frac {1}{y^2}$
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