I am studying this book on differential equations and the author just introduced a new notation I was having trouble trying to understand. I'm not sure if I'm not getting the notation or the concepts themselves.
From what I understand, the exact differential equations are those that have the form
$$M(x,y)+N(x,y)\frac{dy}{dx}=0$$
In which
$$M(x,y)=\frac{\partial F(x,y)}{\partial x} , N(x,y)=\frac{\partial F(x,y)}{\partial y}$$
for some function $F$, that we don't necessarily know. And the partial derivatives of $M$ and $N$ have to be equal. But then they say a differential equation can be written as
$$M(x,y)dx+N(x,y)dy=0$$
What do "$dx$" and "$dy$" mean? And where did the
$$\frac{dy}{dx}$$
go to?
And then they go on to use this notation on every exercise. When you use this notation is it because you already know the equation you're writing down is an exact differential equation?
Then in another part they say:
"the differential
$$dF=F_xdx+F_ydy$$
of F(x,y) is exactly
$$Mdx+Ndy$$
So what does "$dF$" mean?
Best Answer
For your purposes $Mdx+Ndy=0$ is just another notation for the pair $(M,N)$ of functions. In more geometric terms is denotes the direction field that is orthogonal to $(M(x,y),N(x,y))$ at each point $(x,y)$. Solutions are curves that are tangent to the direction field in every point. You can find such solutions by solving the scalar equation $M+Ny'=0$ or by solving a system $$ \dot x(t)=-N(x(t),y(t)),\\ \dot y(t)=M(x(t),y(t)). $$ You can also add some non-zero factor to this system.
Of course the specific notation is chosen for its connection to differential or Pfaffian forms, but these do not offer any further insights at the level of an introductory ODE course.
See also