MAS222: Differential Equations

Semester 1: Ordinary Differential Equations

Lecturer: Jonathan Potts (j.potts@Sheffield.ac.uk)Website: http://jonathan-potts.staff.shef.ac.uk/mas222.html

Chapter 1. Course information• MAS222 is a level 2 course over 2 semesters. The first semester consists of lectures

(two per week) and tutorial classes (1 per week). You are required to attend all of these.

• This is a techniques course, so requires practice. For this, we have provided tutorial sheets and 3 assignment sheets. In the lectures, I will explain when you should be ready to answer these.

• Note: for those who have already downloaded tutorial sheets 2-9, these may change slightly. I’ve now taken the links down. So please re-download them as they re-appear.

• Course website: http://jonathan-potts.staff.shef.ac.uk/mas222.html, containing lecture notes, tutorial sheets, assignment sheets, slides etc.

• Please bring the lecture notes with you to all the lectures. There are empty boxes throughout the notes that need to be filled in either (i) with information written on the board in the lectures, or (ii) given to you as exercises during the lectures.

• My email: j.potts@sheffield.ac.uk• I am away weeks 3 and 4, so Alex Best will be covering my lectures then.

Chapter 2. Introduction to

Ordinary Differential Equations (ODEs)

Introduction to Ordinary Differential Equations (ODEs)

• A differential equation relates a function and its derivatives• Examples:

Introduction to Ordinary Differential Equations (ODEs)

• A differential equation relates a function and its derivatives• Examples:

Introduction to Ordinary Differential Equations (ODEs)

• A differential equation relates a function and its derivatives• Examples:

Here, u is the dependent variable (the variable being differentiated)

x and t are independent variables (the variables we are differentiating by)

Introduction to Ordinary Differential Equations (ODEs)

• A differential equation relates a function and its derivatives• Examples:

Here, u is the dependent variable (the variable being differentiated)

x and t are independent variables (the variables we are differentiating by)

One independent variable => equation is an ODE (subject of semester 1)

More than one => equation is a Partial differential equation (PDE) (semester 2)

Derivative as the gradient function of a curve

Nb. The symbol represents a function of , so can be evaluated at any point . Once evaluated, it is a (real) number. It is very important in mathematics to be aware of what type of object each symbol represents.

Modelling with ODEs• Example 1. A population of organisms.

Modelling with ODEs• Example 1. A population of organisms.• First attempt at a model:

Modelling with ODEs• Example 1. A population of organisms.• First attempt at a model:

• Second attempt:

Modelling with ODEs• Example 1. A population of organisms.• First attempt at a model:

• Second attempt:

• Third attempt:

Solving ODEs: initial conditions

• Frequently, exact solutions are impossible• But sometimes they are easy to find• E.g. the simple population model • Solution: • This is called a general solution. But what is ?• Need some more information about the system: e.g. an initial

condition• Suppose the initial population is of 100 individuals: .• Then . This is a particular solution (particular to this initial condition)

Initial conditions and chaos

• In the previous example, a small change in initial conditions only leads to a small change in the behaviour of the system• If then • If then • Right-hand plot has

…but this is not always the case…

• Consider the following system:

Initial conditions and chaos

Classification of ODEs

• To figure out the best techniques for understanding a particular ODE, it is important to be able to classify it: i.e. ask yourself “what sort of ODE do I have here?”• Four ways of classifying ODEs are by • Order: the order of the highest derivative• Degree: the power to which the highest derivative is raised, after rationalisation• Linearity: the ODE is a sum of terms linear in the independent variable or its derivatives• Homogeneity: if the ODE is linear, it can be written as follows:

Then it is homogeneous if and inhomogeneous otherwise

Solving ODEs: Linear, first order, homogeneous• Such equations look like this: • The general solution is

• Example 5. A population of animals with constant death rate and fluctuating birth rate :

• Solution.

Solving ODEs: Linear, first order, homogeneous

• Qualitative behaviour is governed by the sign of • Left-hand plot shows an example where , and the right-hand where

Solving ODEs: Linear, first order, inhomogeneous• Such equations look like this: • The general solution is

• The term is called the integrating factor• Example 6. Find the solution of the following ODE, with initial

condition :

• Solution.

Separable first order ODEs (possibly non-linear)

• These are often the simplest type of ODE to solve• They are of the form: • This rearranges to • If you can integrate both sides, then you’re done• But beware! Integration is a Dark Art. Many objects, even ones that

seem relatively innocuous, are difficult or impossible to integrate exactly • If you don’t believe me, try calculating . Not so easy, is it?

Separable first order ODEs: ExampleNewton's Law of Cooling states that the rate of change of a body's temperature is proportional to the difference between the temperature of the body and the ambient temperature (assumed constant). You may recall meeting this in MAS110.i. Write down a differential equation representing the law, and find its

general solution as a function of a body's initial temperature (at time ).

ii. Describe the solution in some detail.iii. Calculate how long it takes for the temperature of the body to

reduce to half its initial value in the case that .

Find the solution to the following population model (Example 1):

where . Explain how the long-term behaviour of the solution depends on the parameters , and .

The long term behaviour is as follows:(i) If (i.e. ), then as .(ii) If (i.e. ), then as .

In both cases, determines the speed of approach to the final value

Separable first order ODEs: Example

You should now be able to do tutorial

sheet 1

Chapter 3. Qualitative analysis

of ODEs

Qualitative analysis of ODEs

• Many systems of ODEs are too difficult to solve exactly• However, we can gain much insight by examining qualitative

behaviour, i.e. things like:• What happens to the solution as the dependent variable tends to infinity?

Does it tend to infinity too, or to some constant value? • How does the answer depend on initial or boundary conditions?• Do small perturbations in the system affect the outcome in a “big” way?

• E.g. recall Example 5 from your notes:

• : :

Direction fields

• Your first example of a graphical method for qualitative understanding of ODEs• Suppose • Pick a point and draw a vector with gradient • Repeat for many different pairs

• Example. Newton’s law of cooling:

Direction fields: Newton’s law of cooling

𝑇 𝑎

Direction fields: another example

• Consider the following equation for population growth with seasonally fluctuating births

Autonomous equations and phase lines

• An autonomous ODE is one that does not depend explicitly on the independent variable, e.g. • We can obtain qualitative information of first order autonomous ODEs

by drawing the phase line, as follows• Find the values where (called the equilibrium points)• For the line segments between adjacent equilibrium points, determine the

sign of • Draw a line with equilibrium points marked on it, and arrows to represent the

sign of between these points

Phase line example• Draw the phase line of • Zeros are where .• means that • means that • means that • means that • This means that is a stable equilibrium, and are unstable equilibria• i.e. if the initial condition is close to then , but this is not true for

You should now be able to do tutorial

sheet 2

Chapter 4. Planar, first order,

autonomous systems of ODEs

Definition

• A planar system of ODEs is one where there are two dependent variables• If each of the ODEs is first order and autonomous, the system is

known as a planar, first order, autonomous system and has the following form

• Notice that the system at time can be described by the position of the point , which lies on a plane, hence the word “planar”

Example of a planar system

• Recall Newton’s second law of motion ( stands for “mass” and for “force”):

• Now write • Then we can express the above second order ODE as a planar system

of first order ODEs

Another example of a planar system

• The simple pendulum can be modelled as follows

where is the mass of the pendulum bob, is the length of the pendulum rod and is the angle of deviation from the vertical• Write and • Then we can express the above second order ODE as a planar system

of first order ODEs

The simple pendulum

Above image from https://en.wikipedia.org/wiki/Phase_portrait

Video: https://vimeo.com/53710539

Nullclines

• A full phase portrait can be a lot of work to plot• But much information can be gained by examining the nullclines of

the system

• Nullclines are where or , so represent respectively where or is stationary

Example 12

• A simple model for the concentration of a specific self-regulating protein in a cell is given by

,

• Here, represents the concentration of protein and the concentration of mRNA• If you don’t know about cell transcription, look up the “central dogma

of molecular biology” on the internet. There are some really nice videos like https://www.youtube.com/watch?v=9kOGOY7vthk

Example 12• Nullclines:

• Look at the signs of and in each of the four regions• The blue line is the -nullcline and red is

the -nullcline

Example 12• Nullclines:

• Look at the signs of and in each of the four regions• The blue line is the -nullcline and red is

the -nullcline

(right)

(left)

(down) (up)

Example 12• Nullclines:

• Look at the signs of and in each of the four regions• The blue line is the -nullcline and red is

the -nullcline• The intersection of these two lines is an

equilibrium point

(right)

(left)

(down) (up)

You should now be able to do tutorial

sheet 3