Second-Order Linear Differential Equations
This chapter covers equations of the form
\[y'' + p(t)\,y' + q(t)\,y = g(t),\]
which describe mechanical vibrations, electric circuits and many other systems where an acceleration depends on position and velocity. Following Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems (11th ed., Wiley), Chapter 3, the independent variable is now called \(t\).
Prerequisite: First-Order Differential Equations.
Chapter contents
| Section | What it covers | |
|---|---|---|
| 3.1 | Homogeneous equations with constant coefficients | Linear second-order equations, initial value problems, the characteristic equation, distinct real roots, long-term behaviour |
| 3.2 | Solutions of linear homogeneous equations; the Wronskian | coming soon |
| 3.3 | Complex roots of the characteristic equation | coming soon |
| 3.4 | Repeated roots; reduction of order | coming soon |
| 3.5 | Nonhomogeneous equations; method of undetermined coefficients | coming soon |
| 3.6 | Variation of parameters | coming soon |
| 3.7 | Mechanical and electrical vibrations | coming soon |
| 3.8 | Forced periodic vibrations | coming soon |
Numerical methods for second-order equations (rewriting them as first-order systems) are in Numerical Methods, Section 8.