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:

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

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

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

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

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

Let us look at the following example differential equation:

$$\left(y^{2}-2x\right)dx + \left(2xy+1\right)dy=0$$

Taking the partial derivatives of the functions corresponding to $M\left(x,y\right)$ and $N\left(x,y\right)$, 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 $M\left(x,y\right)$ and $N\left(x,y\right)$,

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

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

similarly for $N\left(x,y\right)$,

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

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:

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

So we have identified a function, $f\left(x,y\right)$, 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.

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