MST209 + MST326 as a 90 point option ?

It seems that MST209 is merely a ( level 2 ) introductory course on differential equations and that MST325 Mathematical Methods and fluid mechanics is the real deal on differential equations as far as the Open University is concerned.

Let me repeat the names of the courses once more:
MST209 - Mathematical methods and models
MST325 - Mathematical Methods and fluid mechanics
Just so that we agree that the word differential equations does not appear in either of the course names. - There is nothing wrong with changing a course description so that it will have appeal to a wider audience as long as the topic remains clear. I.e. "ODEs and PDEs" ( a name I can definitely imagine if the an occasional nerd slipped into the math community ) to "Ordinary and Partial Differential Equations", or: "Differential Equations" to "Mathematical modeling with Differential Equations". I do not understand the strange naming conventions of the Open University.

Names are just names lets have an in-depth look at the course descriptions.

MST209 - What you will study

This course will be of particular interest to you if you use mathematics or mathematical reasoning in your work and feel that you need a firmer grounding in it, or if you think you might find it useful to extend your application of mathematics to a wider range of problems. The course should also be suitable if you are teaching A-level applied mathematics, or if you intend to do so; the material on mechanics, in particular, gives a very careful treatment of the basic concepts of this subject. The teaching is supported and enhanced by the computer algebra package Mathcad.

Around half of this course is about using mathematical models to represent suitable aspects of the real world; the other half is about mathematical methods that are useful in working with such models. The work on models is devoted mainly to the study of classical mechanics, although non-mechanical models – such as those used in heat transfer and population dynamics – are also studied. The work on methods comprises topics chosen for their usefulness in dealing with the models; the main emphasis is on solving the problems arising in the real world, rather than on axiom systems or rigorous proofs. These methods include differential equations, linear algebra, advanced calculus and numerical methods. Many are implemented in Mathcad, so you can use the computer to solve more difficult problems and to investigate case studies.

The mechanics part of the course begins with statics, where there are forces but no motion, and then introduces the fundamental laws governing the motions of bodies acted on by forces – Newton's laws of motion. These are first applied to model the motion of a particle moving in a straight line under the influence of known forces. Undamped oscillations are discussed next. Newton's laws are then extended to the motion of a particle in space. The motions of systems of particles are modelled. Next we look at the damped and forced vibrations of a single particle. Then we look at the motion (and vibrations) of several particles. Finally, we investigate the motion of rigid bodies.

The methods part of the course covers both analytic and numerical methods. The analytical (as opposed to numerical) solution of first-order and of linear, constant-coefficient, second-order ordinary differential equations is discussed, followed by systems of linear and non-linear differential equations and an introduction to methods for solving partial differential equations. The topics in algebra are vector algebra, the theory of matrices and determinants, and eigenvalues and eigenvectors. We develop the elements of the calculus of functions of several variables, including vector calculus and multiple integrals, and make a start on the study of Fourier analysis. Finally, the study of numerical techniques covers the solution of systems of linear algebraic equations, methods for finding eigenvalues and eigenvectors of matrices, and methods for approximating the solution of differential equations.

MST326 - What you will study

In simple terms, we think of a fluid as a substance that flows. Familiar examples are air (a gas) and water (a liquid). All fluids are liquids or gases. The analysis of the forces in and motion of liquids and gases is called fluid mechanics. This course introduces the fundamentals of fluid mechanics and discusses the solutions of fluid-flow problems that are modelled by differential equations. The mathematical methods arise from (and are interpreted in) the context of fluid-flow problems, although they can also be applied in other areas such as electromagnetism and the mechanics of solids.

Because of its many applications, fluid mechanics is important for applied mathematicians, scientists and engineers. The flow of air over objects is of fundamental importance to the aerodynamicist in the design of aeroplanes and to the motor industry in the design of cars with drag-reducing profiles. The flow of fluids through pipes and channels is also important to engineers. Fluid mechanics is essential to the meteorologist in studying the complicated flow patterns in the atmosphere.

The course is arranged in 13 units within four blocks.

Block 1 is the foundation on which the rest of the course is built.

Unit 1 Properties of a fluid introduces the continuum model and many of the properties of a fluid, such as density, pressure and viscosity. The basic equation of fluid statics is formulated and used to find the pressure distribution in a liquid and to provide a model for the atmosphere.

Unit 2 Ordinary differential equations starts by showing how changes of variables (involving use of the Chain Rule) can be applied to solve certain non-constant-coefficient differential equations, and leads on to the topics of boundary-value and eigenvalue problems. It concludes with an introduction to the method of power-series for solving initial-value problems.

Unit 3 First-order partial differential equations extends the earlier version of the Chain Rule to cover a change of variables for functions of two variables, and shows how this leads to the method of characteristics for solving first-order partial differential equations.

Unit 4 Vector field theory relates line, surface and volume integrals through two important theorems – Gauss’ theorem and Stokes’ theorem – and formulates the equation of mass continuity for a fluid in motion.

Block 2 starts by investigating the motion of a fluid that is assumed to be incompressible (its volume cannot be reduced) and inviscid (there is no internal friction).

Unit 5 Kinematics of fluids introduces the equations of streamlines and pathlines, develops the concept of a stream function as a method of describing fluid flows, and formulates Euler’s equation of motion for an inviscid fluid.

Unit 6 Bernoulli’s equation analyses an important equation arising from integrals of Euler’s equation for the flow of an inviscid fluid. It relates pressure, speed and potential energy, and is presented in various forms. Bernoulli’s equation is used to investigate phenomena such as flows through pipes and apertures, through channels and over weirs.

Unit 7 Vorticity discusses two important mathematical tools for modelling fluid flow, the vorticity vector (describing local angular velocity) and circulation. The effects of viscosity on the flow of a real (viscous) fluid past an obstacle are described.

Unit 8 The flow of a viscous fluid establishes the Navier-Stokes equations of motion for a viscous fluid, and investigates some of their exact solutions and some of the simplifications that can be made by applying dimensional arguments.

Block 3 looks at a class of differential equations typified by the wave equation, the diffusion equation and Laplace’s equation, which arise frequently in fluid mechanics and in other branches of applied mathematics.

Unit 9 Second-order partial differential equations shows how a second-order partial differential equation can be classified as one of three standard types, and how to reduce an equation to its standard form. Some general solutions (including d’Alembert’s solution to the wave equation) are found.

Unit 10 Fourier series reviews and develops an important method of approximating a function. The early sections refer to trigonometric Fourier series, and it is shown how these series, together with separation of variables, can be used to represent the solutions of initial-boundary value problems involving the diffusion equation and the wave equation. Later sections generalise to the Fourier series that arise from Sturm-Liouville problems (eigenvalue problems with the differential equation put into a certain standard format), including Legendre series.

Unit 11 Laplace’s equation is a particular second-order partial differential equation that can be used to model the flow of an irrotational, inviscid fluid past a rigid boundary. Solutions to Laplace’s equation are found and interpreted in the context of fluid flow problems, for example, the flow of a fluid past a cylinder and past a sphere.

Block 4 returns to applications of the mathematics to fluid flows.

Unit 12 Water waves uses some of the theory developed in Block 3 to investigate various types of water wave, and discusses several practical examples of these waves.

Unit 13 Boundary layers and turbulence looks at the effects of turbulence (chaotic fluid flow) and at the nature of boundary layers within a flow, introducing models to describe these phenomena.

Now let's have a look at MIT 18.03, the introductory undergraduate course on differential equations at MIT:

18.03 - Description

This course is a study of Ordinary Differential Equations (ODE's), including modeling physical systems.

Topics include:
Solution of First-order ODE's by Analytical, Graphical and Numerical Methods;
Linear ODE's, Especially Second Order with Constant Coefficients;
Undetermined Coefficients and Variation of Parameters;
Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance;
Complex Numbers and Exponentials;
Fourier Series, Periodic Solutions;
Delta Functions, Convolution, and Laplace Transform Methods;
Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

I would say that, roughly, MST209 + MST326 = 18.03, but this means that I must add MST209 + MST326 to my options for 2011. I would not have seen this option if I hadn't been studying MST209 ( OpenLearn ) and the 18.03 lectures.

Difference 18.03 - MST209 ( OpenLearn edition )

Taken from MST209 (OpenLearn edition) Book 2: Second order differential equations:





The 18.03 method however is as follows:

$2\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=2e^{-x}$

$(2D^2-2D+1)y=2e^{-x}$,
( where $\alpha=-1$ and $p(D)=2D^2-2D+1$ )

$y_p = \frac{2}{p(\alpha)}e^{-x} = \frac{2}{5}e^{-x}$

I don't know if the linear differential operator ($D$) is discussed in any part of MST209, if not the conclusion is simple: MIT 18.03 does a better job at teaching differential equations than MST209 since Matuck said that he $D$ operator plays an important role in the rest of 18.03 ( lectures 14-33 ).

Watched MIT 18.03 - lecture 13

I watched another lecture on differential equations by Prof. Arthur Matuck.



Lecture 13 is about finding solutions to the following ODE:
$y'' + Ay' + By = e^{\alpha x}$, with $\alpha$ a complex number.

The general solution is
$y = C_1 y_1 + C_2 y_2 + \frac{e^{\alpha x}}
{\alpha^2 +A\alpha + B }$

where
$C_1 y_1 + C_2 y_2 = y_h$ is a solution the homogeneous part of the ODE and

$\frac{e^{\alpha x}}{\alpha^2 +A\alpha +B} = y_p$ is a particular solution of the ODE.


( I have not blogged about lectures 10,11 and 12 although I have seen them. Lectures 11 and 12 in particular were highly theoretical but did not add much in terms of new definitions or theorems.)

[Fact] - Wroskian of two functions

The Wroskian of two functions $f(x),g(x)$ is
$\left| \begin{array}{cc}
f(x) & g(x)\\
f'(x) & g'(x)
\end{array} \right| = f(x) \cdot g'(x) - f'(x) \cdot g(x)$.

(From MIT 18.03 lecture 11.)

Watched MIT 18.03 - lecture 9

I skipped lectures 7,8 (watched 7 partially and had a brief look at 8 ) because during this first serious confrontation with differential equations I want to follow the route of MST209 ( OpenLearn version ).

In 18.03 lecture 9 Prof. Mattuck talks about differential equations of type $y'' + Ay' + By = 0$. They are of the 2nd order, have constant coefficients and are homogeneous. The procedure for solving them is surprisingly similar to solving second order recurrence equations. The DE has a characteristic equation $r^2 + Ar + B=0$. If both roots are real and distinct the general solution then looks like $y=c_1 \cdot e^{r_1x} + c_2 \cdot e^{r_2x}$. The other two cases ( i.e. a pair of complex conjugates, two real equal roots ) are discussed during the rest of the lecture.

Case: solving linear ODE with trig RHS

In this case we solve the differential equation $y' + 3y = \sin{x} + \cos{x}$. Note the extensive usage of complex numbers.

\[\begin{aligned}
y' + 3y &= \sin{x} + \cos{x}\\
e^{3x} \cdot y' + 3 e^{3x} \cdot y &= e^{3x}(\sin{x} + \cos{x}) \\
(e^{3x} \cdot y)' &= e^{3x}(\sin{x} + \cos{x})\\
(e^{3x} \cdot y)' &= e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})}\\
(e^{3x} \cdot y)' &= \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}}\\
\int{(e^{3x} \cdot y)' \ dx} &= \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}}\\
\int{(e^{3x} \cdot y)' \ dx} &= \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}}\\
e^{3x} \cdot y &= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C\\
y &= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}\\
y &= \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} + Ce^{-3x}
\end{aligned} \]

Step 1:
$y' + 3y = \sin{x} + \cos{x} \Leftrightarrow$
$e^{3x} \cdot y' + 3 e^{3x} \cdot y = e^{3x}(\sin{x} + \cos{x}) $
We want to multiply the LHS and RHS with a factor $p(x)$ such that $p(x) \cdot y + 3 p(x) \cdot y' = (y\cdot p(x))'$. This implies that $p'(x) = 3p(x) \Leftrightarrow p(x) = \int{3p(x) \ dx} \Leftrightarrow p(x) = e^{3x} + C$. For our purpose $C=0$ suffices.


Step 2:
$e^{3x} \cdot y' + 3 e^{3x} \cdot y = e^{3x}(\sin{x} + \cos{x}) \Leftrightarrow$
$(e^{3x} \cdot y)' = e^{3x}(\sin{x} + \cos{x})$
We implement our objective from the previous step. This step is allowed due to the product rule of differentiation: $(f\cdot g)'(x) = f(x)g'(x) + g(x)f'(x)$.


Step 3:
$(e^{3x} \cdot y)' = e^{3x}(\sin{x} + \cos{x}) \Leftrightarrow$
$(e^{3x} \cdot y)' = e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})}$
We want the RHS to be a single trigonometric function:

\[\begin{aligned}
\cos{x}+\sin{x} &= \Re{(e^{ix})} + \Re{(-ie^{ix})}\\
&=\Re{(e^{ix}-ie^{ix})}\\
&=\Re{((1-i)e^{ix})}\\
&=\Re{(\sqrt{2}e^{-\frac{\pi}{4}i} \cdot e^{ix})}\\
&=\Re{(\sqrt{2}e^{(x-\frac{\pi}{4})i})}\\
&=\Re{(\sqrt{2}(\cos{(x-\frac{\pi}{4})}+i\sin{(x-\frac{\pi}{4})}))}\\
&=\sqrt{2}\cos{(x-\frac{\pi}{4})}
\end{aligned} \]


Step 4:
$(e^{3x} \cdot y)' = e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})} \Leftrightarrow$
$(e^{3x} \cdot y)' = \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}}$
If we write the RHS again as the real part of a complex expression we can integrate a single exponential function which is preferable due to its simplicity.


Step 5:
$(e^{3x} \cdot y)' = \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}} \Leftrightarrow$
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}}$
An intermediate step before integration, the RHS needs tyding up in the next step before integration over x.


Step 6:
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}} \Leftrightarrow$
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}}$
In this step we have rewritten the RHS so that it is clear how the integration has to be done.


Step 7:
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}} \Leftrightarrow$
$e^{3x} \cdot y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C$
In this step we have performed a straightforward integration of the LHS and RHS.


Step 8:
$e^{3x} \cdot y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C \Leftrightarrow$
$y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}$
We divide LHS and RHS by $e^{3x}$.


Step 9:
$y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x} \Leftrightarrow$
$y = \frac{2}{5}\sin{x} + \frac{1}{5}\cos{x} + Ce^{-3x}$
We rearrange the RHS as follows:

\[\begin{aligned}
y
&= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}\\
&= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x} \\
&= \Re{\left\{ \frac{1}{3+i} \frac{3-i}{3-i}(1-i)e^x \right\}} + Ce^{-3x} \\
&= \Re{\left\{ (\frac{1}{5}-\frac{2}{5}i)(\cos{x}+i\sin{x}) \right\}} + Ce^{-3x} \\
&= \Re{\left\{ \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} +i(-\frac{2}{5}\cos{x} + \frac{1}{5}\sin{x}) \right\}} + Ce^{-3x} \\
&= \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} + Ce^{-3x}
\end{aligned} \]

This completes the explanation.

Khan Academy

I was listening to one of those great talks by Lew Rockwell. He talked about some 530 million dollar school building in Los Angeles built to house 4,000 (!) high school students and which looked like a prison. Probably because it is a prison for these kids, he added. Lew mentioned that teaching like that is not at all like it should be with the ( technological ) possibilities we have today. Or if you like, with our ( read: the US's ) financial situation. He pointed out that it can be done differently. That it in fact -is- done differently by The Khan Academy, a not-for-profit 501(c)(3) with the mission of providing a world-class education to anyone, anywhere.

For me too? I thought, immediately. And yes, he recently added video lectures for a course in Differential Equations to the approximately 1800 video's already stored on their website. http://www.khanacademy.org/ Besides differential equations they also have an impressive set of calculus video's.

Since I already started the 18.03 lecture program ( competition breeds innovation ! ) I probably continue to do so but I'll be making comparisons.

Logistic Equation

Watched lecture 5 in the MIT 18.03 series


In this lecture differential equations were discussed of type $\frac{dy}{dx}=f(y)$, i.e. with no independent variable in the RHS of the equation, aka autonomous differential equations. He showed how to get qualitative information about the solution without actually solving the Differential Equation. The logistic equation ( remember MST121? ) was discussed and used as a model in several problems. Without actually solving the equation much could be said about the solutions.

I read Lesson 1 of MST209/OpenLearn. The title is 'First Order Differential Equations' and contains procedures for Euler's Method, Direct Integration, Separation of variables and the Integrating Factor method. The lesson also contains several exercises with answers. All very much like a normal Open University lesson.

Watched MIT 18.03 videos 4.

I watched video 4 of MIT 18.03 again today, this time in its entirety.


In this video it is shown how Bernoulli equations and Homogeneous DEs' can be solved by using the right form of variable substitution. Differential Equations is a lot like Calculus in the sense that you have to practice a lot.

Bernouilli equations

I watched approximately half of video lecture 4 of 18.03 today. Topic of lecture 4 is solving differential equations by direct or indirect substitution.

Direct and indirect substitution originates from Calculus Integration, I suppose. Take for example, the integral:
$$\int{x \sin(x^2)\ dx}$$
requires the direct substitutions $y=x^2, dy=2x \ dx$ to solve. However, the integral
$$\int{\frac{1}{\sqrt{1-x^2}}\ dx}$$
requires the indirect substitution $x=sin(u), dx=cos(u) \ du$ to solve.

A Bernoulli equation is a DE of type:
$$y' = p(x) \cdot y + q(x) \cdot y^{n}$$
Rearrange as follows:
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{y}{y^{n}} + q(x) \cdot \frac{y^{n}}{y^{n}}$$
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{1}{y^{n-1}} + q(x) $$
Now substitute $v=\frac{1}{y^{n-1}}$, and thus $v' = (1-n) \frac{y'}{y^{n}}$:
$$\frac{v'}{1-n} = p(x) \cdot v + q(x)$$
$$v' + (n-1)p(x) \cdot v = (1-n)q(x) $$
The last equation is a linear ODE in standard form.

Recipe for solving linear first order ODE

As I blogged before I am going to make more notes, summaries and stuff and store everything in a plex in a Personal Brain database. The following note basically summarizes 18.03 lecture 3.

$$a(x) \cdot y' + b(x) \cdot y = c(x)$$
$$y' + p(x) \cdot y = q(x)$$
$$e^{\int{p(x) \ dx}} \cdot y' + p(x) e^{\int{p(x) \ dx}} \cdot y = e^{\int{p(x) \ dx}} q(x) $$
$$( e^{\int{p(x) \ dx}} \cdot y )'= e^{\int{p(x) \ dx}} q(x) $$
$$e^{\int{p(x) \ dx}} \cdot y= \int{e^{\int{p(x) \ dx}} q(x) \ dx} $$
$$y= \frac{\int{e^{\int{p(x) \ dx}} q(x) \ dx}}{e^{\int{p(x) \ dx}}} + \frac{C}{e^{\int{p(x) \ dx}}}$$

( Would need some explanations but as a personal note it is clear to me. )

Watched MIT 18.03 videos 2 and 3.

Video 3 is about a straightforward recipe for solving a linear ODE of the first order:


Video 2 is on numerically solving DE's with Euler's method:

Getting started on MST209 Differential Equations

In order to get a feeling for MST209 I will be watching the video's of MIT 18.03 Differential Equations by Prof. Arthur Mattuck. I watched lecture 1 today. Since there are 33 video lectures in total I doubt if I'll get through all of them by february though. The purpose is getting back into differential equation stuff. OU OpenLearn has several booklets from the MST209 course available which give a reasonable idea on what to expect of MST209. Both MIT 18.03 and MST209 are less formal than M208, they are mainstream courses teaching mainly how-to's, i.e. how-to solve a particular type of differential equation. - My choice for MST209 next year is more or less certain by now. Three options remain. 1) do just MST209 and concentrate and spend time on self-study of topics that really interest me (i.e. Abstract Algebra stuff ). 2) Add M381 Number Theory + Logic. 3) Add M337 Complex Analysis. Well, I haven't decided yet. MST209 involves 7 TMA's, 2CMA's and probably a MS221, M208 'type-of-exam', i.e. 12 questions for 70, plus 2 out of 5 for 30.