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 2? 2
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