Question

Which of the following is the solution to the differentiable equation dy/dx=4x/y, where Y(2)=-2

Answer 1

First we'll substitute $\frac{dy}{dx}$ with $y'$

$y'=\frac{4x}{y}$

Then we can separate this.

$yy'=4x$

Then we'll solve this.

$yy'=4x: \frac{y^2}{2}=2x^2+c_1$

$\frac{y^2}{2}=2x^2-6$

$y=2\sqrt{x^2-3},y=-2\sqrt{x^2-3}$

Then we'll plug in to find the extraneous solutions (if any)

$y=-2\sqrt{x^2-3}$