Theory

Paths in the plane

A classical set of applications of first-order ODEs involves finding paths through space with specific properties.

Paths in the plane (part 2)

Catenary vs. tractrix
The catenary curve is the evolute of the tractrix; the tractrix is the involute of the catenary.
https://en.wikipedia.org/wiki/Involute#Involute_and_evolute

Growth and decay

Another classic application of differential equations produces exponential growth and decay. Variants on this theme can account for recurring additions and subtractions.

Autonomous equations

In a first calculus class, the sign of the derivative $y^{\prime }$ of a function $y(x)$ is written on the number line in order to indicate places where $y$ is increasing or decreasing. This technique is useful for determining whether a critical point is a local min, max, or neither.

The same technique is used to study first-order autonomous differential equations. Autonomous equations having the form $\frac{dy}{dt}=f(y)$, meaning that the defining function $f$ does not involve the independent variable $t$.

The main differences are: (i) for autonomous ODEs we have the derivative $f(y)$ but we do not already have the function $y(t)$, and (ii) for autonomous ODEs we draw the signs of $f(y)$ as vertical arrows (up or down) on a vertical line, instead of “$+$” and “$-$” on a horizontal line. We can add filled dots where $f(y)=0$, and open dots or dashed lines where $f(y)=\pm \infty$, since these lines have special meaning.

This vertical number line we use to analyze $\frac{dy}{dt}=f(y)$ is called a phase line. Phase lines can be used to make qualitative observations about the behavior of solutions to autonomous ODEs.

Lines $y=y_0$ corresponding to zeros of $f(y)$ yield flat solutions. If the derivative is negative at such a line, i.e. $f^{\prime }(y_0)<0$, so $f(y)$ cross through zero from above to below, then solutions will be attracted to the line $y=y_0$. These lines are called stable equilibria. If $f^{\prime }(y_0)>0$ then solutions will be repelled from the line. These lines are called unstable equilibria. (Sometimes stable equilibria are called sinks and unstable ones which are repulsive on both sides are called sources, while unstable ones that are repulsive on one side and attractive on the other side are called nodes.)

When $f^{\prime }(y_0)=0$, the equilibrium $y=y_0$ could be stable, unstable, or neither, and additional information is needed. The situation is the same as for critical points of a function that fails the second-derivative test.

Phase lines:

Phase lines with $f\to \infty$ at $y=1$:

Phase lines deduced from the graph of $f(y)$ alone:

Logistic growth
Many natural systems experience exponential growth and decay only to first order. Once the exponential behavior has occurred for significant time, environmental effects of the growth or decay begin to modulate the rate of that growth or decay.

Exponential growth satisfies the equation $y^{\prime }(t)=ry$, where $r$ is a constant giving the growth rate. We can modify the equation to account for changes of growth rate that depend on the quantity $y$ by letting $r$ become a function $h(y)$.