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this is a question that's had me sort of stumped. Any points or help would be greatly appreciated, I've never been that great at differential equations

The question at hand

The presence of the alpha seems to throw off any actual working I do such as manipulating for a value of y prime, and I'd rather use an integration factor but the change of variables has been specified for? I also can't seem to find any change of variable questions similar enough to follow the working. Thanks in advance to whoever helps out.

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$$y'-2xy=\dfrac {x^3}{y^3}$$ This is Bernoulli's differential equation. $$y^3y'-2xy^4=x^3$$ Note that $(y^4)'=4y^3y'$. Substitute $v=y^4$ and solve.

You can surely take it from here.

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  • $\begingroup$ Ah! Completely missed that by just trying to take it head on, but that makes so much sense. Thank you! $\endgroup$ Jan 29, 2021 at 11:06
  • $\begingroup$ You're welcome. The differential equation is now linear of first order. Thats easy to solve. Also take a look at the wiki page on the Bernoulli differential equation. @AlexStephens $\endgroup$ Jan 29, 2021 at 11:07

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