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Calculus I / Ordinary Differential Equations

Overview

A differential equation expresses a relationship between an unknown function and its derivatives. In finance, ODEs govern everything from the continuous compounding formula dB/dt=rBdB/dt = rB (solved by B(t)=B0ertB(t) = B_0 e^{rt}) to the Ornstein-Uhlenbeck mean-reversion dynamics dx=a(b−x) dtdx = a(b-x)\,dt that underlie interest rate and commodity models. This topic develops the two main solution techniques for first-order ODEs — separation of variables and the integrating factor — and applies them to exponential growth and decay, logistic population models, and linear mean-reversion.

By the end of this topic you should be able to:

  • read a direction field and identify equilibrium solutions;
  • solve separable equations with initial conditions;
  • solve linear first-order ODEs using the integrating factor μ(x)=e∫P(x) dx\mu(x) = e^{\int P(x)\,dx};
  • derive the logistic equation and solve it via separation of variables;
  • solve the OU mean-reversion ODE and interpret aa, bb in financial terms.

In this lesson

  1. Contents
  2. The big picture

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