I've been introduced to a new concept called the integrating factor and I unsure how it effects the separation of variables for this equation. Can anyone help...
By multiplying the following differential equation by $2ye^x$ and carefully considering the resulting expression, show that $2ye^x$ is an integrating factor for $$\frac{dy}{dx} + \frac{2x+ x^2 + y^2}{2y}=0$$ and find an explicit expression, $y=y(x)$, for the general solution of the equation.
Thanks in advance
I wouldn't think of it in terms of separation of variables -- in fact, I'd think of separation of variables as a special case of the notion of exactness of a differential equation. Exactness doesn't come for free, but integrating factors are precisely the things that provides exactness. For the current example, among other notational possibilities, I would multiple the expression by $2ye^x$ and re-write as $$ (2x+x^2+y^2)e^x\,dx+2ye^x\,dy=0. $$ The point is that the left-hand side is now of the form $dF$ for the function $$ F(x,y)=e^x(x^2+y^2), $$ and the solutions to $dF=0$ are given by $F(x,y)=c$ for an arbitrary real constant $c$. We can solve $e^x(x^2+y^2)=c$ explicitly for $y$ to get $$\boxed{y=\pm\sqrt{ce^{-x}-x^2}.}$$ In general, you might only be able get $y$ implicitly in terms of $x$, but that's just one of those things you have to get used to.
