MTH2032 Differential Equations with Modelling
(Semester 2, 2022)
Assignment 1
(due Wednesday 24 August 2022, 6pm)
Exercise 1 (25 marks). The following ODE
dy
dx
=
?x+√x2 + y2
y
describes the shape of a plane curve that will reflect all incoming light beams to the same point
and could be a model for the mirror of a reflecting telescope, a statellite antenna or a solar
collector.
1. [5 marks] Verify that the ODE is homogeneous type.
2. [10 marks] Solve the ODE by substituting u = y
x
.
3. [10 marks] Show that the ODE can also be solved by means of the substitution u =
x2 + y2.
Exercise 2 (25 marks). We consider the third order differential equation
y′′′ =
√
1 + (y′′)2
couple with initial conditions y(0) = 1, y′(0) = ?1, y′′(0) = 1
1. [5 marks] Reduce the IVP to a first order system of IVPs.
2. [10 marks] Show that the IVP has a unique solution.
3. [10 marks] Solve the IVP by subtituting u = y′′ to reduce the order of the ODE.
Exercise 3 (15 marks). We consider the ODE
mx2 cos(y)? xm sin(y)y′ = 0
1. [5 marks] Find values of m such that the ODE is exact.
2. [10 marks] Solve the ODE with values of m that you found in question 1.
Exercise 4 (10 marks). Find all initial conditions such that the ODE
(x2 ? x)y′ = (2x? 1)y
has no solution, precisely one solution and more than one solution.
1
Exercise 5 (25 marks). Let α and β be real numbers and consider the following numerical
method to approximate the solutions to the IVP y′ = f(y) with initial condition y(0) = y0:
starting from y0, for all n ≥ 0 define yn+1 by
y?n+1 = yn +
2h
3
f(yn) (first predictor)
y??n+1 = yn +
h
3
f(yn) (second predictor)
yn+1 = yn + h
[
αf
(
y?n+1
)
+ βf
(
y??n+1
)]
(corrector).
1. [10 marks] The quantities y?n+1 and y
n+1 predict the values of the solution y at certain
points in the interval [xn, xn + h]. Which ones? Justify your answer.
2. [5 marks] Find a function Φ(y, h) such that the method can be written yn+1 = yn +
hΦ(yn, h).
3. [10 marks] We assume that f is indefinitely differentiable with continuous derivatives.
For which conditions of α and β has the method a truncation error of order 1? Of order