I have one problem in this task and please help me solve it. The problem is a first order non linear equation: $$xy \frac{dy}{dx}=1-x^2.$$ Here I have moved $y$ in one side and $x$ to the other, so I have
$$y\frac{dy}{dx}=\frac{1}{x}-x.$$
Here, I integrate both sides. On the right, I get:
$$\ln(x)-\frac{x^2}{2}+c$$
but what about left part? $\displaystyle \int y \frac{dy}{dx}$
How do I evaluate that? I have tried to take $y$ as a function of $x$, for example $y=kx$, $\frac{dy}{dx}=k$
and so $y \frac{dy}{dx}=kxk=k^2x$ ; if we integrate we get $k^2x^2/2=y^2/2$ so does it mean that $\displaystyle \int y \frac{dy}{dx} =\frac{y^2}{2}$ ?
thanks