Chapter 9

The Laplace Transform

The Laplace transform converts a linear differential equation with constant coefficients into an algebraic equation. Derivatives become multiplication by $s$ , initial conditions enter at the first step, and switched or impulsive inputs, which are awkward for the methods of Chapter 4, are handled as easily as smooth ones. It is the everyday language of circuit analysis, vibrations and control engineering, where transfer functions, poles and impulse responses all come from this chapter.

The chapter has five sections:

9.1 Definition and properties: the transform as an improper integral, when it exists, linearity, a table of transforms, shifting in $s$ , transforms of derivatives, multiplication by $t^{n}$ and periodic functions.
9.2 Inverse transform and convolution: getting back from $F (s)$ to $f (t)$ with partial fractions and completing the square, and the convolution theorem.
9.3 Solving initial-value problems: the complete method for constant-coefficient equations of first, second and third order.
9.4 Discontinuous forcing: unit step functions, the second shifting theorem, pulses, the Dirac delta function and the impulse response.
9.5 Linear systems: two coupled equations solved with Cramer's rule in the $s$ -domain.

What you will be able to do

Compute Laplace transforms from the definition and, much faster, from a table and a few rules.
Invert rational transforms by partial fractions, including repeated and quadratic factors.
Solve linear initial-value problems in one pass, without finding a general solution first.
Model on/off inputs with step functions and sudden blows with the delta function, and solve the resulting problems.
Solve small linear systems with initial conditions.

Prerequisites

You need integration by parts, improper integrals and limits, partial fractions from calculus, and $2 \times 2$ determinants. From earlier chapters: constant-coefficient linear equations and their characteristic polynomial (Sections 4.2–4.3); for Section 9.5, the idea of a linear system (Chapter 7) is helpful but not required.