The theory of differential equations developed alongside the invention of calculus itself, driven by the demands of physics — particularly celestial mechanics, vibrating strings, and heat conduction. The subject cannot be attributed to a single founder; instead, it emerged from the intertwined contributions of Newton, Leibniz, the Bernoulli family, Euler, and their successors over roughly two centuries, from the 1670s to the 1870s.

**Origins in the calculus of Newton and Leibniz (1670s–1690s).** Newton's method of fluxions (developed from the mid-1660s but not published until later) treated rates of change as fundamental objects, and many of Newton's calculations in the *Principia* (1687) amount to solving what we would now call separable first-order ODEs — for instance, his analysis of resistance forces on moving bodies. Leibniz, working independently from the 1670s, introduced the notation $dy/dx$ and the integral sign $\int$ that we still use, and was the first to explicitly pose the problem of finding a curve $y(x)$ satisfying a given relation between $x$, $y$, and $dy/dx$. The term "differential equation" is due to Leibniz (1676).

**The Bernoulli era and exact equations (1690s–1740s).** Jakob Bernoulli and his brother Johann made differential equations into a systematic discipline. Jakob introduced the method of separation of variables (1690) in his work on the catenary problem, and Johann developed the first techniques for solving exact equations and for finding integrating factors. The famous brachistochrone problem (1696) — posed by Johann as a challenge to the mathematical community — was solved independently by Newton, Leibniz, Jakob Bernoulli, and L'Hôpital, and its solution required reducing a variational problem to an ODE. Johann Bernoulli also introduced the method of variation of parameters in embryonic form, though the general technique would not be fully developed until Lagrange's work several decades later.

**Euler's systematisation (1740s–1770s).** Leonhard Euler transformed the subject from a collection of ad hoc tricks into a coherent theory. His *Institutionum Calculi Integralis* (1768–1770) was the first textbook on differential equations and remains remarkably modern in its organisation. Euler introduced the exponential ansatz $y = e^{\lambda x}$ for constant-coefficient linear ODEs, derived the characteristic equation, and gave the complete classification into real distinct roots, repeated roots, and complex conjugate roots that we teach today. He also initiated the study of special functions through the ODE lens: Bessel's equation, the hypergeometric equation, and Euler's own equidimensional equation all appear in his work (though Bessel functions are named after Friedrich Bessel, who studied them in the context of planetary perturbations in the 1820s). Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ — essential for converting complex exponentials into real trigonometric solutions — first appeared in his *Introductio in Analysin Infinitorum* (1748).

**Lagrange and variation of parameters (1770s–1780s).** Joseph-Louis Lagrange gave the first fully general statement and proof of the variation of parameters method (1774–1775), showing that it reduces any inhomogeneous linear ODE to quadratures (integrals) once the complementary function is known. Lagrange also contributed the first rigorous treatment of the Wronskian determinant's role in testing linear independence, although the name "Wronskian" was coined later by Thomas Muir (1882) in honour of the Polish mathematician Józef Hoene-Wroński, who studied such determinants in the early 19th century.

**The 19th century: existence, uniqueness, and series solutions (1800s–1870s).** The shift from finding explicit solutions to proving that solutions *exist* began with Cauchy's existence and uniqueness theorem (1820s–1840s), which established that $dy/dx = f(x, y)$ with $f$ Lipschitz continuous has a unique solution through each initial point. This was a foundational moment: it meant that qualitative analysis — studying the behaviour of solutions without finding them explicitly — could be placed on rigorous footing. Lazarus Fuchs (1866) classified the singular points of linear ODEs and proved the theorem that bears his name, guaranteeing convergent Frobenius series solutions at regular singular points. Ferdinand Georg Frobenius (1873) refined Fuchs's method into the systematic procedure taught today, including the indicial equation and the three cases for the second solution (non-integer difference, integer difference, and coincident exponents).

**Stability theory and dynamical systems (1880s–1900s).** The qualitative theory of differential equations — studying the global behaviour of solutions rather than their explicit form — was initiated by Henri Poincaré in his monumental *Mémoire sur les courbes définies par une équation différentielle* (1881–1886). Poincaré introduced the concepts of phase portraits, limit cycles, and the classification of equilibria into nodes, saddles, spirals, and centres that form the core of Chapter 5 of these notes. His work was motivated by the three-body problem in celestial mechanics, where explicit solutions are impossible and qualitative methods are the only recourse. Aleksandr Lyapunov (1892) placed the stability analysis on rigorous foundations with his doctoral thesis *The General Problem of the Stability of Motion*, introducing the linearisation method and the direct method (Lyapunov functions) that remain the standard tools today.

**D'Alembert and the wave equation (1747).** Jean le Rond d'Alembert's solution of the one-dimensional wave equation — published in 1747, predating much of the ODE theory above — was one of the earliest applications of partial differential equations to physics. D'Alembert's argument (introducing the change of variables $\xi = x + ct$, $\eta = x - ct$ to factorise the wave operator) is essentially the one presented in Chapter 5, and his explicit formula $u = F(\xi) + G(\eta)$ remains the starting point for the modern theory of hyperbolic PDEs. The subsequent debate between d'Alembert, Euler, and Daniel Bernoulli over the meaning of "function" and the legitimacy of Fourier series solutions — Could a function with corners satisfy the wave equation? Must solutions be expressible as trigonometric series? — drove the development of rigorous analysis throughout the 18th and 19th centuries and ultimately led to the modern theories of distributions and Sobolev spaces.