MTH2032代写、代写Java,python编程语言

日期：2022-08-28 11:11

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