Second- and Higher-Order Differential Equations

Start with the notes

Begin with the introductory note on second-order linear differential equations. It introduces the form of a second-order ordinary differential equation and explains how its order relates to derivatives. Use it to establish the language before moving on to a method.

Next, read Method of Undetermined Coefficients. It sets out steps for finding a complementary function and a particular integral. The note on power series solutions offers another place to continue, starting with a review of power and Taylor series.

Use the maps to connect ideas

Nth Order ODEs brings together standard forms, constant coefficients and variation of parameters. Compare it with the mind map on second-order ordinary differential equations, which covers homogeneous equations and initial values. Looking between the maps can help you connect the second-order material to the higher-order forms shown on this page.

Another mind map covers the characteristic equation and general solutions for homogeneous second-order equations. A map on power series solutions includes the Legendre equation and second-order equations with singular points. Use these alongside the notes when you want a different view of the material.

Practice and check terms

After reading, try FP3 differential equations. Its questions include finding an integrating factor and a complementary function. Return to the relevant note or map as you work through the questions, then use the quiz again to practice the steps.

The flashcards on second-order ordinary differential equations include boundary conditions, the Wronskian and the superposition principle. Use them to go over terms that appear in the material.

Explore the unit

This page is one section of the Differential Equations unit in Mathematics. Its notes, maps, quiz and flashcards offer different ways to study the material gathered under Second- and Higher-Order Differential Equations. Choose a resource that fits what you want to practice next.