Separable but not exact equation

Question

In class, my professor stated that all separable equations are exact, and we even proved it for homework, but I think I found an equation that is separable but not exact:

$$(x\ln y+xy)+(y\ln x+xy)y′ =0$$

My Work:

\begin{align*} M &= x\ln y + xy \\ M_y &= \frac{x}{y} +x \\ N &= y\ln x + xy \\ N_x &= \frac{y}{x} +y \\ M_y &\neq N_x \end{align*}

But \begin{align*} x(\ln y + y) + y \times y'\times (\ln x +x) &= 0 \\ x(\ln y + y) &= - y \times y'\times (\ln x +x) \\ \frac{x}{\ln x +x} &= \frac{- y \times y'}{\ln y + y} \end{align*}

Separated

So whats up? Am I doing something wrong?

Answer 1

The correct statement is that a separable differential equation in the form

$$ a(x)\; dx + b(y)\; dy = 0$$

is exact. But if you multiply an exact differential equation by some function of $x$ and/or $y$, it generally will cease to be exact. So, for example, $x a(x)\; dx + x b(y)\; dy = 0$ will not be exact, though it is still separable.