How to find the general solution of this equation?

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$$y'=(xy'+y)y^3$$I don't know how to approach this problem with the two $y'$.

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2 Answers

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Notice that $xy' + y = (xy)'$, which means that $\dfrac{y'}{y^3} = (xy)'$. Can you continue?

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Move all $y'$ terms to one side:$$\begin{align}y'&=(xy'+y)y^3\\y'&=xy^3y'+y^4\\y'-xy^3y'&=y^4\\y'(1-xy^3)&=y^4\\y'&=\frac{y^4}{1-xy^3}\end{align}$$Does this get you closer to a form that you know how to solve?

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