Differential Equations

Prerequisites: Calculus.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A differential equation is a rule that tells us how something changes, rather than what it is. "The population grows by 2 percent per year" or "the acceleration of the ball is proportional to the force on it" are statements of this kind. The unknown is a whole function, for example the position of the ball at every moment, and the equation links the function to its own rates of change.

Such a rule alone does not fix the answer. A ball thrown upward and a ball dropped from a tower obey the same law of motion but follow different paths. To get one definite answer we also need a starting point: where the ball is and how fast it moves at the beginning. Given the rule and the starting point, we can step forward in time. From the current state the rule gives the current rate of change, which tells us the state a moment later, and so on.

This is the form in which most laws of physics are written. Newton's laws, Maxwell's equations and the Schrödinger equation all state how things change from one moment to the next or from one point to its neighbours. Solving a differential equation means turning such a local rule into a prediction for the whole history of the system.

Concrete

A differential equation relates an unknown function to its derivatives. If the function depends on one variable, it is an ordinary differential equation (ODE); if it depends on several variables and partial derivatives appear, it is a partial differential equation (PDE). The order is the highest derivative that appears. The equation is linear if the unknown and its derivatives appear only to the first power and are not multiplied with each other.

First order. Radioactive decay with rate $\lambda > 0$ obeys

$$\frac{dN}{dt} = -\lambda N.$$

Separating variables, $dN/N = -\lambda\, dt$, and integrating from $0$ to $t$ gives $N(t) = N(0)\, e^{-\lambda t}$. One integration constant appears, fixed by the initial value $N(0)$.

Second order. A mass $m$ on a spring with constant $k$ obeys Newton's law $m\ddot x = -kx$, or

$$\ddot x + \omega^2 x = 0, \qquad \omega = \sqrt{k/m}.$$

Trying $x = e^{rt}$ gives $r^2 + \omega^2 = 0$, so $r = \pm i\omega$, and the real general solution is

$$x(t) = A\cos\omega t + B\sin\omega t.$$

A second-order equation needs two initial conditions; here $x(0) = A$ and $\dot x(0) = B\omega$.

Superposition. For a linear homogeneous equation $L x = 0$, any linear combination of solutions is again a solution. The general solution of the inhomogeneous equation $Lx = F$ is one particular solution plus the general solution of $Lx = 0$. For the driven oscillator $\ddot x + \omega^2 x = F_0 \cos \Omega t$ with $\Omega \neq \omega$, the ansatz $x_p = C\cos\Omega t$ gives $C = F_0/(\omega^2 - \Omega^2)$, so

$$x(t) = A\cos\omega t + B\sin\omega t + \frac{F_0}{\omega^2 - \Omega^2}\cos\Omega t.$$

Partial differential equations. The standard examples in physics are second-order and linear:

$$\begin{aligned} \frac{\partial^2 u}{\partial t^2} &= c^2 \nabla^2 u &&\text{(wave equation)}, \\ \frac{\partial u}{\partial t} &= D \nabla^2 u &&\text{(diffusion or heat equation)}, \\ \nabla^2 \phi &= -\rho/\varepsilon_0 &&\text{(Poisson equation)}. \end{aligned}$$

For a PDE the integration "constants" become functions, so we need initial data on a whole region (for example $u$ and $\partial_t u$ at $t = 0$ for the wave equation) and boundary conditions. A common method is separation of variables: one writes $u(t, \vec x) = T(t) X(\vec x)$, which turns a linear PDE with suitable symmetry into several ODEs. The Fourier transform is the other standard tool for linear equations with constant coefficients, since it turns derivatives into multiplication.

• Linear Differential Operators by Lanczos - as all his books, useful and full of unusual ways to look at the subject.
• TEN LESSONS I WISH I HAD LEARNED BEFORE I STARTED TEACHING DIFFERENTIAL EQUATIONS GIAN-CARLO ROTA

Abstract

ODEs as vector fields. Any explicit ODE system (solved for the highest derivative) can be written in first-order form $\dot x = f(t, x)$ with $x \in \mathbb{R}^n$, by introducing derivatives as new variables (for $m\ddot q = F(q)$ take $x = (q, p)$ with $\dot q = p/m$, $\dot p = F(q)$). Geometrically, an autonomous system $\dot x = f(x)$ on a manifold $M$ is a vector field $f$, and a solution is an integral curve of it. The Picard-Lindelöf theorem states: if $f$ is continuous in $t$ and locally Lipschitz in $x$, then for every initial value $x(t_0) = x_0$ there is a unique solution on some interval around $t_0$. The solutions define a local flow $\phi_t: M \to M$. In Hamiltonian mechanics this flow is generated by the Hamiltonian vector field on phase space and preserves the symplectic form.

For a linear system $\dot x = A x$ with constant matrix $A$ the flow is $\phi_t = e^{tA}$, the matrix exponential. For time-dependent $A(t)$ the solution operator is a time-ordered exponential.

Linear PDEs. A linear differential operator of order $m$ on $\mathbb{R}^n$ is

$$L = \sum_{|\alpha| \leq m} a_\alpha(x)\, \partial^\alpha,$$

with multi-indices $\alpha$. Its principal symbol $\sigma_L(x, \xi) = \sum_{|\alpha| = m} a_\alpha(x)\, (i\xi)^\alpha$ governs the qualitative behaviour. For second-order operators $a^{ij}(x)\partial_i \partial_j + \dots$ with real symmetric $a^{ij}$, one calls $L$ elliptic where $a^{ij}$ is definite (Laplace, Poisson), hyperbolic where it has signature $(1, n-1)$ or $(n-1, 1)$ (wave equation), and parabolic in the degenerate case typical of the heat equation $\partial_t - D\nabla^2$, whose second-order part has rank $n - 1$ in the variables $(t, \vec x)$. Elliptic equations describe equilibria and come with boundary value problems. Hyperbolic equations describe propagation with finite speed and come with initial value problems on spacelike surfaces.

Fundamental solutions. A fundamental solution (Green's function) of $L$ is a distribution $G$ with $LG = \delta$. Then $u = G * F$ solves $Lu = F$. For the Laplacian in three dimensions,

$$\nabla^2 \left(-\frac{1}{4\pi |\vec x|}\right) = \delta^3(\vec x).$$

The Malgrange-Ehrenpreis theorem guarantees that every nonzero linear differential operator with constant coefficients has a fundamental solution. In general one looks for weak solutions in Sobolev spaces, where derivatives are understood in the distributional sense. Nonlinear PDEs, such as the Navier-Stokes or Einstein equations, have no such general theory, and existence and uniqueness are studied case by case.

Why is it interesting?

Many of the most important equations in physics are differential equations.

Leads to: Boundary Conditions, Harmonic Oscillator.