Differential Eqns @diff_eq
Tweets on ordinary and partial differential equations from @JohnDCook johndcook.com/blog/partial-d… Joined March 2013-
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A cautionary tale about numerical solutions to differential equations: johndcook.com/blog/2009/08/1…
Reduction of order. Suppose you have one solution y_1 to y'' + p(x)y' + qy = 0. Let y_2 = u y_1. Leads to a first order ODE for u.
There's a $1M prize for proving a theorem about the Navier-Stokes equations. claymath.org/millennium/nav…
If y is a vector-valued function and A is a matrix, the solution to y' = Ay is exp(At)y(0).
Laplace transforms are more useful than an intro class may imply. johndcook.com/blog/2024/10/2…
To solve a homogeneous ODE y' = f(y/x), introduce u = y/x and replace y’ with u+xu’ to turn the ODE into a separable ODE in terms of x and u.
The term 'homogeneous' has two meanings in ODEs: linear equations with no forcing term, and equations of the form y' = f(y/x).
Solving Laplace's equation in the upper half plane johndcook.com/blog/2022/11/2…
“The three-body problem exhibits almost all of the intractability of the more general N-body problem.” — Michael Spivak
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.
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