MATH2121
THEORY AND APPLICATIONS OF DIFFERENTIAL
EQUATIONS
Term 2, 2019, Class Test 2 V1
Time Allowed: 45 mins
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This sheet must be filled in and attached to the front of your answers.
An approved calculator is allowed.
See over for questions
A Bessel Formula
d
dx
(xνJν(x)) = x
νJν−1(x)
1. [13 marks] Consider the ODE
3x2y′′ + 2xy′ + x2y = 0
(a) Show that it has a a regular singular point at x = 0.
(b) Apply Frobenius method about x = 0 to determine the indicial
equation and its roots.
(c) Determine the recurrence relation.
(d) Find the first 3 non-zero terms of the series solution corresponding
to the larger root.
2. [5 marks]
(a) Find the general solution to
x2y′′ + xy′ + (2x2 − 4)y = 0
(b) Evaluate ∫
x5J2(x) dx
3. [8 marks] Consider the ODE,
y′′ − 2xy′ + 8y = 0.
(a) Find the recurrence relation from seeking power series solutions
about x = 0.
(b) Explain from the recurrence relation that one of the series solution
terminates and becomes a polynomial.
(c) Find the polynomial solution.
4. [7 marks] Find all the critical points (equilibrium solutions) of
dx
dt
= 2x− x2 − xy
dy
dt
= 3y − 2y2 − 3xy

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