Tweets on ordinary and partial differential equations from @JohnDCook
Joined March 2013
- Tweets 4,632
- Following 17
- Followers 75,806
- Likes 31
291 Photos and videos
Jun 11
'Knowing what is big and what is small is more important than being able to solve partial differential equations.' -- Stan Ulam
2
4
34
2,437
Jun 10
Laplace transforms turn differential equations into algebraic equations.
2
10
60
2,808
Math developed for going to the moon johndcook.com/blog/2020/06/1…
10
1,407
To solve y'' p(x) y' q(x) y = 0, assume a power series for y, plug it in, and solve recurrence for coefficients.
1
5
27
2,494
We usually start with a DE and then look for solutions.
But it can be useful to find a DE that a function of interest satisfies.
If it’s a nice DE some properties of the solutions follow easily.
2
4
18
2,160
Laguerre polynomials satisfy x y'' (n 1-x) y' k y = 0. These polynomials come up on quantum mechanics.
1
21
1,692
'The central problem of ... modern mathematics is the study of ... functions defined by differential equations.' -- Felix Klein
5
22
2,009
A cautionary tale about numerical solutions to differential equations: johndcook.com/blog/2009/08/1…
1
22
1,865
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.
11
1,686
There's a $1M prize for proving a theorem about the Navier-Stokes equations. claymath.org/millennium/navi…
2
4
14
1,976
May 29
If y is a vector-valued function and A is a matrix, the solution to y' = Ay is exp(At)y(0).
6
33
2,410
May 28
Laplace transforms are more useful than an intro class may imply. johndcook.com/blog/2024/10/2…
3
27
1,772
May 27
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.
4
18
1,636
May 26
The term 'homogeneous' has two meanings in ODEs: linear equations with no forcing term, and equations of the form y' = f(y/x).
2
22
1,990
May 22
“The three-body problem exhibits almost all of the intractability of the more general N-body problem.” — Michael Spivak
4
9
51
8,483
May 21
An equation of the form M(x) N(y)y' = 0 is called separable. Write as M dx = - N dy and integrate both sides.
4
28
3,239
May 19
You can usually apply initial conditions as an afterthought, but boundary conditions have to be baked in from the beginning.
12
68
3,252