houghton@maths.tcd.ie

**Laplace Transforms**

**Z Transforms**

**Problem Sheet 6**: pdf or ps. Solutions pdf, small pdf, ps or small ps. prob6.tex and soln6.tex.
Taking the Z-transform, the shift theorems.

**Problem Sheet 7**: pdf or ps. Solutions pdf, small pdf, ps or small ps. prob7.tex and soln7.tex.
Solving difference equations.

**Systems of differential equations**

**Problem Sheet 8**: pdf or ps. Solutions pdf, small pdf, ps or small ps. prob8.tex and soln8.tex, tanks.eps (a eps of tanks for the mixing problem). Revising eigenvectors. Mixing problems.

**Problem Sheet 9**: pdf or ps. Solutions pdf, small pdf, ps or small ps. prob9.tex and soln9.tex.
Revising eigenvectors. Solving linear systems.

**Problem Sheet 10**: pdf or ps. Solutions pdf, small pdf, ps or small ps.
prob10.tex and soln10.tar (tar file including eps of nodes).
Phase diagrams and stationary points.

**Problem Sheet 11**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob11.tex and soln11.tar (tar file including eps of nodes).
More phase diagrams and stationary points.

**Problem Sheet 12**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob12.tex and soln12.tar (tar file including eps of nodes).
Drawing spirals, only one eigenvector.

**Problem Sheet 13**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob13.tex and soln13.tex. Inhomogeneous equations.

**Linearization**

**Problem Sheet 14**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob14.tex and soln14.tar (tar file including eps of phase diagram).
Linearization.

**Problem Sheet 15**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob15.tex and soln15.tar (tar file including eps of phase diagram).
More linearization.

**Series solutions**

**Problem Sheet 16**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob16.tex and soln16.tex.
Series solutions.

**Problem Sheet 17**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob17.tex and soln17.tex.
More series solutions, method of Frobenius.

**Problem Sheet 18**: Question 1 is about Legendre polynomials.

**Vector calculus**

**Problem Sheet 18**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob18.tex and soln18.tex.
Legendre polynomials, revision of vectors.

**Problem Sheet 19**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob19.tex and soln19.tex, 16April2001.eps (an eps showing (sin^3t,cos^3t)), curve11apr2001.epsi (an eps showing a corkscrew spiral).
Revision of vectors, lengths of curves.

**Problem Sheet 20**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob20.tex and soln20.tex The gradient.

**Problem Sheet 21**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob21.tex and soln21.tex The div and curl.

**Problem Sheet 22**: pdf
or ps. Solutions
pdf,
small
pdf, ps
or small
ps.
prob22.tex and soln22.tex The Gauss theorem.

Table of Z-transforms. pdf or ps. Ztnew.tex

Notes on Z-transforms, part 1. pdf, ps, short pdf or short ps. note1.tex.

Notes on Z-transforms, part 2. pdf, ps, short pdf or short ps.
note2.tex.

Notes on Z-transforms, part 3. pdf, ps, short pdf or short ps.
note3.tex.

Notes on Z-transforms, part 4. pdf, ps, short pdf or short ps.
note4.tex.

Table of Laplace transforms. pdf or ps laplacet-03-04.tex

Laplace transform of cosh*(at)*cosh*(bt)*. pdf or ps. 12Oct2001.tex This is an example of using the first shift theorem.

A long example of solving a second
order problem using Laplace transforms. pdf, short pdf, ps or short ps
19Oct2001.tex

Examples with complex roots, pdf, short pdf, ps or short ps 6Nov2002.tar includes epsi plot of solution.

Simple example with Heaviside function. pdf
or ps 6Nov2001.tex

Another Heaviside function example and a Dirac delta function example, pdf
or ps or short pdf
or short ps. 18Nov2003.tex

The convolution cos*t**cos*t*. pdf
or ps.

Another convolution example. pdf or ps. 12Nov2002.tex

The Laplace transform of a periodic function. pdf
or ps.

The Laplace transform of a saw tooth function. pdf, ps, short pdf or short ps. soln5.tex.