MATH2121
THEORY AND APPLICATIONS OF DIFFERENTIAL
EQUATIONS
Online T2, 2020
Class Test 2 Version 1
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A Bessel Formula
d
dx
(xνJν(x)) = x
νJν−1(x)
1. [13 marks] Consider the ODE
3x2y′′ + 2xy′ + x2y = 0
The Frobenius method is used to find the solution about x0 = 0.
(a) Show the ODE has a regular singular point at x0 = 0.
(b) Determine the indicial equation and its roots.
(c) Determine the recurrence relation.
(d) Write out the power series solution corresponding to the larger
root showing explicitly the first 3 non-zero terms.
2. [5 marks]
(a) Find the general solution to the ODEs
i.
x2y′′ + xy′ + (2x2 − 1)y = 0
ii.
x2y′′ + xy′ + 36x2y = 0
(b) Evaluate ∫
x5J2(x) dx
3. [8 marks] We use the power series method to solve this equation about
the ordinary point x0 = 0,
y′′ − 2xy′ + 8y = 0.
(a) Find the recurrence relation.
(b) Explain from the recurrence relation that one of the series solution
terminates and becomes a polynomial.
(c) Write down the polynomial solution.
4. [9 marks] Consider the following linear system
d
dt
[
x
y
]
=
[
2 1
1 −3
] [
x
y
]
(a) Determine the type and stability of the equilibrium point (0, 0)T
(b) Find the general solution.
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