Ordinary Differential Equations (17): Physics and Engineering Applications

Fri, 29 Mar 2024 09:00:00 +0000

Differential equations are not a pure mathematical game — they are the language for understanding the physical world. From celestial motion to circuit response, from a swinging pendulum to vortex shedding behind a bridge cable, every dynamical system “speaks” ODE.

This chapter is a deliberate tour through five canonical applications. Each one will pay back the entire ODE toolkit we built in chapters 1-16: phase planes, eigenvalues, Laplace transforms, modal analysis, conservation laws, numerical integration, control. None of the examples is a “toy” — they are all genuine working physics, written tightly so that the structure remains visible.

Ordinary Differential Equations (3): Higher-Order Linear Theory

Fri, 04 Aug 2023 09:00:00 +0000

A first-order ODE has memory of one number; a second-order ODE has memory of two. That tiny extra degree of freedom is what lets the same equation describe a plucked guitar string, the suspension of your car, the L-C tank circuit inside an FM radio, and the swaying of a tall building in the wind. In every case the same three regimes appear — oscillate, return-with-a-touch-of-overshoot, or crawl back — and the same algebraic gadget, the characteristic equation, predicts which one happens.

RLC Circuit on Chen Kai Blog

Ordinary Differential Equations (17): Physics and Engineering Applications

Ordinary Differential Equations (3): Higher-Order Linear Theory