Chapter 4

Higher-Order Linear Equations: Explicit Methods

Most of the differential equations that engineers meet every day are linear and of second or higher order. Vibrating springs and structures, RLC circuits, beams under load and simple control loops all lead to such equations. This chapter is the core of the course. It explains why linear equations behave so well, and it gives you explicit methods that produce exact formulas for their solutions.

The chapter has two halves. The first builds the theory and the main tools:

4.1 Basic theory: existence and uniqueness, superposition, linear independence, the Wronskian, fundamental sets, reduction of order, and the structure $y = y_{c} + y_{p}$ .
4.2 Constant coefficients: the characteristic equation and its three cases (distinct, repeated and complex roots), with Euler's formula and a short guide to finding polynomial roots.
4.3 Undetermined coefficients: a fast way to find particular solutions when the forcing is a polynomial, exponential, sine, cosine or a product of these.

The second half extends these ideas: 4.4 variation of parameters handles any continuous forcing, 4.5 treats the Cauchy–Euler equation, whose coefficients are powers of $x$ , and 4.6 collects deeper theorems on second-order homogeneous equations.

What you will be able to do

Decide on which interval a linear initial-value problem has a unique solution.
Test whether solutions are independent and build the general solution.
Solve any homogeneous linear equation with constant coefficients once the roots of its characteristic polynomial are known.
Find particular solutions and solve nonhomogeneous initial-value problems.

Prerequisites

You should be comfortable with differentiation and integration (including the product rule and integration by parts), with $2 \times 2$ and $3 \times 3$ determinants, and with complex numbers in the form $a + ib$ . From earlier chapters you need the idea of a solution and of an initial-value problem (Chapter 1) and the first-order linear equation (Section 2.3).

What you will be able to do

Prerequisites

Sections