A differential equation is an equation in which the unknown is a function, and the equation speaks of its derivatives. Euler organized this path into a scheme that we still use today: observe a phenomenon, write the equation, find the function that solves it and draw the graph.

Euler's scheme: phenomenon → differential equation → solution function → graph. The derivative also points to the maximum and minimum points.
Euler’s scheme: phenomenon → differential equation → solution function → graph. The derivative also points to the maximum and minimum points.Euler’s Eyes Collection

The phenomenon. In 1678, Robert Hooke summarized the behavior of a spring in Latin: ut tensio, sic vis, “as extension, so strength.” A spring of constant $k$, stretched by $x$, pulls back with strength $F = -kx$.

A body of mass m attached to a spring of constant k.
A body of mass m attached to a spring of constant k.Euler’s Eyes Collection
The question: what is the function x = x(t) that gives the position at each time t?
The question: what is the function x = x(t) that gives the position at each time t?Euler’s Eyes Collection
F = m·a with F = −kx: the differential equation appears.
F = m·a with F = −kx: the differential equation appears.Euler’s Eyes Collection

The equation. By Newton’s second law, $F = m \cdot a$, and acceleration is the second derivative of position, $x''$. Then

$$-kx = m\,x'' \quad\Longrightarrow\quad x'' + \frac{k}{m}\,x = 0.$$

The solution function. We look for a function whose second derivative is itself multiplied by a negative number. Euler’s trick (1743) was to try $x = e^{rt}$, since differentiating $e^{rt}$ only does the factor $r$ appears:

$$r^2 e^{rt} + \frac{k}{m}\,e^{rt} = 0 \quad\Longrightarrow\quad r^2 = -\frac{k}{m} \quad\Longrightarrow\quad r = \pm\, i\,\omega, \quad \omega = \sqrt{\frac{k}{m}}.$$

An imaginary number appeared! This is where Euler’s formula comes in: $e^{i\omega t} = \cos(\omega t) + i\sin(\omega t)$. Putting the real parts together, the solutions are

$$x(t) = c_1 \sin(\omega t) + c_2 \cos(\omega t).$$

The graph. The mass goes back and forth forever, drawing a sinusoid. Change the spring and mass and see what period $T = 2\pi/\omega$ change: harder spring oscillates faster, larger mass oscillates slower.

The mass oscillates and draws x(t). Try k = 16 and m = 1: the period is halved compared to k = 4.
Elastic curves: the same equation with different constants.
Elastic curves: the same equation with different constants.Euler’s Eyes Collection
Page of the *Institutiones Calculi Differentialis* (1755), where Euler organized the differential calculus.
Page of the Institutiones Calculi Differentialis (1755), where Euler organized the differential calculus.Euler’s Eyes Collection

What function is this? (click to see the next puzzle)

  1. There is a function $f$ whose derivative, minus $2x$, gives zero: $f' - 2x = 0$. Response: $f(x) = x^2$ (plus any constant).
  2. There is a function $h$ whose second derivative, added to the function itself, gives zero: $h'' + h = 0$. Response: $h(x) = \sin x$, and also $\cos x$. It is the spring equation with $k = m$.
  3. There is a function $g$ whose fourth derivative, minus the function itself, gives zero: $g'''' - g = 0$. Response: $g(x) = \cos x$, and also $\sin x$, $e^x$ and $e^{-x}$.

All topics by Leonhard Euler · ← Euler’s formula and the “most beautiful equation” · The Königsberg bridges and the birth of graphs →

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