Exact Differential Equations

The general form of a first-order differential equation is given by the following:

$$M(x,y)dx+N(x,y)dy=0$$

our differential equation is said to be exact if it satisfies the following exactness test:

$$\frac{\partial M(x,y)}{\partial y}=\frac{\partial N(x,y)}{\partial x}$$

The goal is to determine a function $f(x,y)$ that satisfies the following:

$$df=M(x,y)dx+N(x,y)dy$$

$$\frac{\partial f(x,y)}{\partial x}=M(x,y)$$

$$\frac{\partial f(x,y)}{\partial y}=N(x,y)$$

Let us look at the following example differential equation:

$$(y^2-2x)dx+(2xy+1)dy=0$$

Taking the partial derivatives of the functions corresponding to $M(x,y)$ and $N(x,y)$, we get:

$$\frac{\partial M}{\partial y}=2y$$

$$\frac{\partial N}{\partial x}=2y$$

So our differential equation is indeed exact and we now can find the total function, $f(x,y)$, whose derivative is equal to our differential equation. This is done by integrating  the functions $M(x,y)$ and $N(x,y)$,

$$M(x,y)=\frac{\partial f(x,y)}{\partial x}$$

$$f=\int (y^2-2x)dx=x{y}^2-x^2$$

similarly for $N(x,y)$,

$$N(x,y)=\frac{\partial f(x,y)}{\partial y}$$
$$f=\int (2xy+1)dy=x{y}^2+y$$

In both cases, we ignore the constant of integration. We now can identify unique terms and construct the function, $f(x,y)$, by summing these terms:

$$f(x,y)=x{y}^2-x^2+y=\text{constant}$$

So we have identified a function, $f(x,y)$, that is a solution to our exact differential equation.

Now for our quote:

I became an atheist because, as a graduate student studying quantum physics, life seemed to be reducible to second-order differential equations. It thus became apparent to me that mathematics, physics, and chemistry had it all and I didn't see any need to go beyond that.
-Attributed to Francis Collins but unconfirmed.

Reuse and Attribution

Bringuier, S., Exact Differential Equations, Dirac's Student, (2019). Retrieved from https://www.diracs-student.blog/2019/04/exact-differential-equations.html.

  @misc{Bringuier_11APR2019,
  title        = {Exact Differential Equations},
  author       = {Bringuier, Stefan},
  year         = 2019,
  month        = apr,
  url          = {https://www.diracs-student.blog/2019/04/}# 
                 {exact-differential-equations.html},
  note         = {Accessed: 2025-05-28},
  howpublished = {Dirac's Student [Blog]},
  }