Mathematical Modeling with Differential Equations

Start with population models

Begin with The Logistic Model, a note that considers population growth when resources are finite. It contrasts that situation with an exponential model that assumes unlimited resources. The note introduces the idea of a maximum population size that can be supported.

Next, read Harvesting. It describes constant harvesting, where a fixed amount is taken regardless of population size, and harvesting where a fixed fraction of the population is removed. Reading the notes together gives you two ways to think about how a population is represented in a model.

Move from a problem to an equation

The mind map Formulating differential equations from word problems starts by identifying independent and dependent variables and assigning letters to them. It then considers units of measurement. Use it after the notes to focus on the steps involved in setting up a model from a written problem.

Modelling in Applied Mathematics introduces dimensional analysis, including mass, length and time, and the need for dimensional consistency. It also includes an exponential model. Look at it alongside the word-problem map to compare the choices involved in setting up an equation.

Explore other models

The mind maps on differential equations and mathematical models include cooling and heating, natural growth and radioactive decay. Higher Equations ODEs Models includes mass-spring-dashpot models and electric circuits. These give you other examples to explore after the population notes.

Use the quizzes and maps together

Try a quiz after studying a map, then return to the map to review ideas relevant to the questions. This topic is one section of the Differential Equations unit in Mathematics, so you can also return to the unit after exploring the resources here. Choose a map and a quiz to get started.