Question

verify that -2x^2y+y^2=1 is the implicit solution of the of the differential equation (x^2-y)dy/dx + 2xy=0

Answer 1

Let us verify that $-2x^2y+y^2=1$ is the implicit solution of the of the differential equation $(x^2-y)\frac{dy}{dx} + 2xy=0.$ For this let us find the implicit derivative of $-2x^2y+y^2=1.$ It follows that

$-4xy-2x^2\frac{dy}{dx}+2y\frac{dy}{dx}=0,$ that is equivalent to $2xy+x^2\frac{dy}{dx}-y\frac{dy}{dx}=0.$ The last equation is equivalent to $(x^2-y)\frac{dy}{dx} + 2xy=0.$ Consequently, $-2x^2y+y^2=1$ is indeed the implicit solution of the of the differential equation $(x^2-y)\frac{dy}{dx} + 2xy=0.$