Differential Equations

What this unit covers

The resources here follow the usual path through the topic. Classification comes first: ordinary or partial, order, linear or nonlinear. Then come first order methods, second order equations, series solutions, Laplace transforms and an introduction to partial equations and dynamical systems.

Where to start

Begin with the mind map Differential Equations. It sets out the classification and gives you the vocabulary the rest of the page relies on. Next, open the mind map First Order ODE. It works as a decision path: check whether the equation is autonomous, integrate both sides, use the initial condition and look for singular solutions.

Once first order work feels familiar, move to the notes on second order linear equations, which begin with the general form y'' = f(t, y, y'). The flashcard deck Linear Differential Equations covers reduction of order, variation of parameters, the method of Frobenius and the D'Alembert formula.

Going further

• Laplace transforms: the mind map Laplace Transform covers the table, s shifting, partial fractions, inversion and solving ODEs. The deck Diff EQ Laplace Transforms Table is handy for quick recall.
• Modelling: the notes The Logistic Model, Linearisation and Harvesting show equations applied to real situations.
• Dynamics: the deck Non Linear Dynamic Systems links linearised systems to stability through eigenvalues.
• Partial equations: the deck B5.2 poses the Cauchy-Kowalevski theorem for first order quasilinear equations.

Using quizzes and maps together

Read a mind map first to see how the methods connect. Then take a quiz such as First Order Differential Equation or First Order Linear Quiz#2 to test whether you can pick the right method. Where you slip, go back to the map branch or flashcards for that method and try again. Teachers can use the maps as ready-made lesson summaries and the quizzes as quick checks.

Where it fits

This unit sits in the Calculus strand of Mathematics, within the Sciences area. Nearly all the material is pitched at university level, with a small share for school. It was made by GoConqr users, so styles and notation vary. Pick one resource, work through it, and then build a map or deck of your own.