A differential equation is a constraint on a function and its derivatives. Wherever a rate of change depends on a current state, the natural model is a differential equation. The continuous-compounding growth equation says "the growth rate of a bank balance is proportional to the current balance" — and its solution is the fundamental formula underlying present-value calculations, bond pricing, and the risk-free discount factor. Before learning to solve ODEs analytically, this lesson shows how to understand their qualitative behaviour geometrically using direction fields — a technique that works even when no closed-form solution exists.
Learning objectives. By the end of this note you should be able to:
- set up a differential equation from a verbal or financial description;
- sketch a direction field for a first-order ODE ;
- identify equilibrium solutions and their stability;
- check a proposed solution by substitution.
In this lesson
- What is a differential equation?
- Building ODE models
- Direction fields
- Equilibrium solutions
- Remark 1 (When direction fields matter)
Before you start
- The Derivative as a Limit: Rates of Change and the Derivative Function
- Exponential Growth, Decay, and Continuous Compounding
The full lesson, with worked examples and exercise solutions, is for members.
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