MTH2032 Differential Equations with Modelling
(Semester 2, 2022)
Assignment 1(due Wednesday 24 August 2022, 6pm)
Exercise 1 (25 marks). The following ODEdy
d=?x+√x2 + y2y
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.
- [5 marks] Verify that the ODE is homogeneous type.
- [10 marks] Solve the ODE by substituting u = y
- [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 equationy′′′ =√1 + (y′′)2couple with initial conditions y(0) = 1, y′(0) = ?1, y′′(0) = 1 - [5 marks] Reduce the IVP to a first order system of IVPs.
- [10 marks] Show that the IVP has a unique solution.
- [10 marks] Solve the IVP by subtituting u = y′′ to reduce the order of the ODE.
Exercise 3 (15 marks). We consider the ODEmx2 cos(y)? xm sin(y)y′ = 0 - [5 marks] Find values of m such that the ODE is exact.
- [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)yhas no solution, precisely one solution and more than one solution.
Exercise 5 (25 marks). Let α and β be real numbers and consider the following numericalmethod 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 byy?n+1 = yn +2h3f(yn) (first predictor)y??n+1 = yn +h
3f(yn) (second predictor)yn+1 = yn + h(corrector). - [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. - [5 marks] Find a function Φ(y, h) such that the method can be written yn+1 = yn +hΦ(yn, h).
- [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
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