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