Differential Eqns

Differential equation modeling johndcook.com/blog/partial-d…

An equation of the form M(x) + N(y)y' = 0 is called separable. Write as M dx = - N dy and integrate both sides.

Boundary conditions are the hard part: johndcook.com/blog/2012/01/2…

You can usually apply initial conditions as an afterthought, but boundary conditions have to be baked in from the beginning.

The essence of chaos johndcook.com/blog/2023/04/2…

'Never in my life have I heard of anyone solving a first order differential equation by finding an integrating factor.' -- Gian-Carlo Rota

It's OK to use dubious math to produce a solution to an ODE as long as you rigorously show that the solution is correct.

Solving Laplace's equation on a disk johndcook.com/blog/2022/11/2…

“If I only knew how I could get mathematicians interested in transformation groups and the treatment of differential equations that arises from them. I am absolutely certain, that, at some point in the future, these theories will be recognized as fundamental.” -- Sophus Lie

NASA's favorite ODE solver johndcook.com/blog/2020/06/1…

'[Richardson] extrapolation was discovered by a hint from theory followed by arithmetical experiments, which gave pleasing results.' -- L. F. Richardson, 1927

To solve y' + p(x) y = g(x), multiply both sides by the integrating factor exp(P(x)) where P'(x) = p(x). Then integrate.

Runge-Kutta methods and Butcher tableau johndcook.com/blog/2020/02/1…

The big three PDEs are Laplace's equation, heat equation, and wave equation. Canonical examples of elliptic, parabolic, and hyperbolic.

Hearing the shape of a triangle ams.org/notices/201311… [pdf]

You can turn an nth order ODE into a system of n first order ODEs by introducing new variables.

The Sobolev embedding theorem: Trading generalized derivatives for classical ones johndcook.com/blog/2022/10/1…

ODE = ordinary differential equation, function of variable. PDE = partial differential equation, function of several variables.

The most common differential equations in application are 2nd order, because Newton's laws are 2nd order.

Oscillatory differential equations johndcook.com/blog/2021/07/0…

Closed-form solution nonlinear pendulum w/ Jacobi functions johndcook.com/blog/2026/04/2…