SCIENCE ONE: MATHEMATICS ASSIGNMENT 14
There are two parts to this assignment. The first part is called Assignment 14 (online); you can find it
at www.mathxl.com. The second part consists of the questions on this page. You are expected to provide full
solutions with complete arguments and justifications. You will be graded primarily on the correctness, clarity
and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled,
with your name and student number at the top of each page.
The Bessel equation is a differential equation which arises in problems of wave propagation. It is given by
x2 y 00 + xy 0 + (x2 − 1)y = 0.
This is a difficult differential equation to solve because the coefficients of y 00 , y 0 and y are not integers.
(You will see in class that differential equations of the form ay 00 + by 0 + c = 0 are fairly tractable.) In this
assignment, you will find one nontrivial solution to the Bessel equation using series. The technical skills —
combining and differentiating series, testing convergence — are all required for the exam. Solving differential
equations using series is an extremely powerful idea, and widely applicable in higher-level courses in physics
and math; but it will not be tested in April.
1. Consider a proposed solution to the Bessel equation of the form
y=
∞
X
an xµ+n .
n=0
Substitute this into the left-hand side of the Bessel equation, and simplify into a single power series.
2. Let a0 = 1 and a1 = 0. Determine a2 , a3 , a4 , . . ..
3. Prove that the series described in question 2 converges.