I'm watching a KhanAcademy video. The question is to solve the exact diff. eq.

$\displaystyle y \cos x + 2xe^y + (\sin x +x^2e^y-1)y^\prime=0$

All of the steps are (obviously) fine and he comes up with

$\displaystyle \psi(x,y)=y \sin x + x^2 e^y - y + C$

which he then uses to say the solution is

$\displaystyle y \sin x + x^2 e^y - y = C$

I understand those steps, too (well, I think I do).But what is that solution? Is that y?The solution to a diff. eq. should be a function, y, whic, when you take its derivatives, satisfies the original equation.

So what was $\displaystyle \psi$? Is it just a $\displaystyle y_{general}(x,y)$ which solved the original? Can I say that [tex]\psi(x,y)=y_g(x,y)[\tex]? Note that if I add the partial of $\displaystyle \psi$ wrt x to the partial wrt y times y', I get the original question back.

I'm not really sure how to ask my question. Does it make sense? Think of this question like a beginning algebra student asking what it means that $\displaystyle x=5$ is the solution to $\displaystyle 2x-10=0$. You answer that question by telling him to plug 5 back into the original equation.)