# Who wrote down the first differential equation?

I am just curious who was the first person to write down a differential equation? And what was this differential equation?

• It might have been Euler. My numerical analysis teacher used to say you may find contribution about any thing in his works! – Medi1Saif Apr 11 '16 at 13:02
• I think Newton and Leibniz allready did use these methods. But I am curious if someone else allready used this type of method, even if not in the modern sense. – MrYouMath Apr 11 '16 at 13:03

It is impossible to tell "who wrote down" first. Leibniz PUBLISHED first (1684), but some documents indicate that he knew how to solve some differential equations in 1666. Newton never published his "Method of fluxions", but it is claimed that he discovered it in 1665 to 1667.

Method of fluxions of Newton was published after his death in 1736 but it was written in 1671. The first differential equation encountered in this book is $$(3x^2-2ax+ay)dx+(ax-3y^2)dy=0$$ which by our modern classification is exact.

But the story is more complicated than this. Newton (and others) actually SOLVED differential equations in many cases without WRITING them. Using some geometric arguments. For example, Newton in fact solved the differential equation which describes the two body problem in celestial mechanics, though he never wrote it as a differential equation, did not use the formal notion of derivative or fluxion, but argued by a purely geometric method. In the last volume of Principia he deals with very advanced questions (perturbation theory) which nowadays belong to the theory of differential equations, but he did not state them in these terms.

The name "differential equation" was introduced by Leibniz. The earliest published paper, of which I could find an English translation, and where a differential equation is written and solved is this:

http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf

He solves the "isochrone problem".

Remark. In fact Galileo solved some simple differential equation long before Newton, when he was trying to figure out the path of a falling body. If $$y(t)$$ is the height at time $$t$$, $$y(0)=0$$, his first guess was that $$y'=ky$$ speed proportional to traveled distance. Then by complicated reasoning we concluded that this cannot be the correct law. In modern language $$y(t)=ce^{kt}$$; if you satisfy the initial condition you obtain $$c=0$$, $$y(t)=0$$, so the motion never begins. Galileo's second guess was the correct $$y'=kt$$, which has a solution $$y(t)=kt^2/2$$, the Galileo law of the falling bodies. Of course, Galileo could not use these simple notations, and had to use some complicated arguments, but his arguments solving these differential equations were correct.

"But this will appear plainer by an Example or two. ..." (Newton 1671) --- After outlining his general method for finding solutions of differential equations.

and

Newton obtained the solution of a differential equation satisfying a given initial condition in terms of infinite series. At each stage of his series solution, he inserted the series into his differential equation and integrated the resulting polynomial.

• What differential equation did he solve? – MrYouMath Apr 11 '16 at 17:34
• The first example is what we now call "Newton's Law of Cooling". – Gerald Edgar Apr 11 '16 at 17:44

A good case could be made for Kepler to be the first to pose and solve a differential equation, although it involved a large amount of trial and error. In particular, his Astronomia Nova (1609) is in its essence a treatise on how he figured out how to go from the physical hypothesis that the Sun attracts planets to a set of (angular) equations that embodies this hypothesis:

$$\dfrac{dE}{dt} \propto \frac{1}{r}\\ \dfrac{dV}{dt} \propto \frac{1}{r^2}$$

where $$t$$ is time, $$r$$ is the distance to the Sun, $$V$$ is the angle about the Sun (called the true anomaly; $$V$$ for latin verus), and $$E$$ is the angle about the center of the orbit (called the eccentric anomaly).

Ultimately, he shows that elliptical motion satisfies this set of equations, a result we now call Kepler's First and Second Laws of Planetary Motion.

## Aside

There is plenty of other mathematical and history of science gold in the book, such as:

1. Proving isomorphism of Ptolemaic, Copernican, and Tychonic systems.
2. Use of first and second order finite differences in analyzing motion.
3. Posing an inverse problem of extreme practical importance for astronomers, but with no closed-form solution. Gauss' attempt to simplify calculations of it led to his first application of hypergeometric series, yielding a highly efficient numerical solution (see his 1809 Theoria motus corporum coelestium...).
• Would you please cite the page or section where you're saying that Kepler wrote these things down? The book is online at archive.org/details/Astronomianovaa00Kepl . I think you'll find he didn't write down equations at all, but gave his information in the form of diagrams, verbal descriptions and numerical examples. – terry-s Dec 16 '18 at 23:21