Exact differential equations conundrum

Question

I'm studying EDE's from 'Advanced Engineering mathematics' by 'Michael Greenberg'. I pretty much understood the procedure adopted to know if there exists a function $F(x,y)$ as a solution to the EDE. However, I fail to understand the assumption of $x$ and $y$ to be independent variables in the beginning of the argument. I mean, in the end we often find an implicit or explicit solution of $x$ and $y$, relating one to the other and yet here we are, "assuming" both the variables to be independent of each other all the way. How does that make any sense? For all I know, the very procedure would fail badly if both the variables are dependent as there would be no meaning then of the partial derivatives in the arguments that follow. So, how does it all make any sense? Isn't it all very contradictory to assume one thing and find just the opposite in the end ?

The argument that would fail if $x$ and $y$ become dependent:

\begin{equation} df=A\,dx+B\,dy=0 \tag{1} \end{equation} Now \begin{equation} \dfrac{df}{dx}=A+B\,\dfrac{dy}{dx} \end{equation}

Therefore $\dfrac{\partial f}{\partial x}=A$ since in the partial differential case $dy=0$.

Similarly, $\dfrac{\partial f}{\partial y}=B$.

Answer 1

The solutions to an exact equation are the level curves of a two-variable function, $z=F(x,y)$. As far as $z$ is concerned, $x$ and $y$ are independent variables. But when you take a level curve of $F$, by setting $F(x,y) = k$ for some constant $k$, you impose a condition on $x$ and $y$ by saying they need to be on this curve. In that case, knowing $x$ tells you something about $y$, but only because you've insisted they be on the same level curve. So $x$ and $y$ are independent, in regards to the surface $z=F(x,y)$ but dependent when you restrict to a level-curve.

Answer 2

When one takes a partial derivative of a function of two or more variables, one holds all variables constant except the variable with respect to which one is taking the partial derivative. To see a visual example consider the function

\begin{equation} f(x,y)=x^2y+y \end{equation}

Suppose we hold $y$ constant at the value $y=1$. The diagram below show the surface graph of $f(x,y)$ as well as the intersection of the surface with the plane $y=1$. As you would expect, the intersection of the two surfaces is a parabola.

When you take $\dfrac{\partial}{\partial x}f(x,y)$ you are finding the slope of a tangent line to that parabolic intersection.

The particular parabola will change is $y$ is set to a different constant value, but the slope of the parabola will always satisfy the equation

\begin{equation} \dfrac{\partial}{\partial x}f(x,y)=2xy \end{equation}

for whichever $(x,y)$.

Answer 3

I think I'm thoroughly confused in this matter now