Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution. $(x^2+y^2)^2=4xy^2$
2026-03-29 06:27:18.1774765638
implicit differential equation, y is a solution
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The technique of implicit differentiation is the key in such problems. So, basically, your first step is to implicit differentiate given equation, and then solve for $\dfrac{\mathrm dy}{\mathrm dx}$.
Given equation: $(x^2+y^2)^2=4xy^2$.
1st step:
$2(x^2+y^2)(2x+2yy')=4y^2+8xyy'$
2nd step:
$4x(x^2+y^2) + 4y(x^2+y^2)y'=4y^2+8xyy'$
$y'\left[ 4y(x^2+y^2)-8xy \right] = 4y^2-4x(x^2+y^2)$
$\boxed{y' = \dfrac{4y^2-4x(x^2+y^2)}{4y(x^2+y^2-2x)}}$.