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.

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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 mass m attached to a spring of constant k; the displacement oscillates between −A and +A around the equilibrium O.
A mass m attached to a spring of constant k; the displacement oscillates between −A and +A around the equilibrium O.Redrawn diagram based on sketch from the Euler’s Eyes collection
Time t enters the system and position x leaves: x = x(t).
Time t enters the system and position x leaves: x = x(t).Original diagram based on the sketches provided
With F = m·a, F = −kx and a = x″, we obtain the differential equation of motion.
With F = m·a, F = −kx and a = x″, we obtain the differential equation of motion.Original diagram based on the sketches provided

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.
Six curves of the general solution; the coefficients and frequency change the shape of the oscillation.
Six curves of the general solution; the coefficients and frequency change the shape of the oscillation.Original diagram, with curves calculated from the general solution
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; redesigned page with transparent background

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 →

Credits and sources of images.