Differential Eqns

Tweets on ordinary and partial differential equations from @JohnDCook

[Differential Eqns @diff_eq]

Differential equation modeling
https://t.co/ePqOKgEpiL

[Differential Eqns @diff_eq]

Third order ODEs https://t.co/5RKnPzot4F

[Differential Eqns @diff_eq]

Third-order ODEs are uncommon, and usually nonlinear, such as the Blasius equation. https://t.co/8JjZmrukx0

[Differential Eqns @diff_eq]

Bernoulli's equation: y' + p(x) y = q(x) y^n. Can be turned into a linear equation by the substitution u = y^(1-n).

[Differential Eqns @diff_eq]

Delay differential equations

https://t.co/CagdhjKG8v

[Differential Eqns @diff_eq]

Dirichlet boundary conditions specify solution values on boundary. Neumann boundary conditions specify normal derivatives of solution.

[Differential Eqns @diff_eq]

There are important non-linear PDEs with closed-form solutions https://t.co/ejGjRodMyj

[Differential Eqns @diff_eq]

The canonical example of elliptical PDEs is Laplace's equation.

[Differential Eqns @diff_eq]

“Good theory is (almost) as useful as exact formulas.” — Lawrence Evans re PDEs

[Differential Eqns @diff_eq]

Conservative vector fields and exact ODEs https://t.co/jIQms6h8KJ

[Differential Eqns @diff_eq]

A solution corresponding to an eigenvalue with positive real part grows exponentially.

[Differential Eqns @diff_eq]

If a linear ODE has complex eigenvalues, solutions oscillate. If real parts are negative, oscillations decay exponentially.

[Differential Eqns @diff_eq]

Blow up in finite time https://t.co/uYyooli9Yc

[Differential Eqns @diff_eq]

'Knowing what is big and what is small is more important than being able to solve partial differential equations.' -- Stan Ulam

[Differential Eqns @diff_eq]

Laplace transforms turn differential equations into algebraic equations.

[Differential Eqns @diff_eq]

Math developed for going to the moon https://t.co/32RZVlBGdT

[Differential Eqns @diff_eq]

To solve y'' + p(x) y' + q(x) y = 0, assume a power series for y, plug it in, and solve recurrence for coefficients.

[Differential Eqns @diff_eq]

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.

[Differential Eqns @diff_eq]

Laguerre polynomials satisfy x y'' + (n+1-x) y' + k y = 0. These polynomials come up on quantum mechanics.

[Differential Eqns @diff_eq]

'The central problem of ... modern mathematics is the study of ... functions defined by differential equations.' -- Felix Klein

[Differential Eqns @diff_eq]

A cautionary tale about numerical solutions to differential equations: https://t.co/U9Ag4ytS3r