Ode to ODEs

· by · in Mathematics, Physics. ·

I always felt a little cheated when I took my Economics and Mathematics degree – because the differential equation work was fairly light touch. Well actually very light touch – to the point I can’t really recall a significant amount of what was covered. The module itself was worth 30 credits: for those au fait with the UK university system, 1/36th of the total degree.

For anyone interested in the world around us, it’s hard not to be lured in by differential equations. They are descriptions of the world in a language that the entire mathematical world can understand. One thing that certainly adds a huge amount of appeal is their history. The equations were invented by Isaac Newton (and Liebniz), which is in itself a pretty impressive fact. It also brings about a debate I have had on many long train rides (with myself) was Mathematics invented or discovered? Attending Trinity College in the mid to late 1600s, the father of calculus realised that these equations could relate functions of real physical things to their rate of change, creating the differential equation. Methodus fluxionum et Serierum Infinitarum lists the following differential equations (very helpfully able to be cut and paste from Wikipedia). [Note: to the average reader stumbling upon a Physics blog in their free time this likely basic knowledge; but do bear with it, I would like to express my appreciation from the start.]

\frac {dy}{dx} = f(x)

\frac {dy}{dx} = f(x,y)

x_1 \frac {\partial y}{\partial x_1} + x_2 \frac {\partial y}{\partial x_2} = y

To me this is just the coolest. All those things you saw in A-level text books and got so familiar with… you forget there was a time these things had never been written down. They didn’t always exist, and here they are at their origin. Anyway enough – I am sure most people have already made their own minds up on differential equations.

I had started my journey on the aforementioned lecture notes from Prof Binney which served me well for complex numbers; but I will hold my hands up and say that they were a little dense for my starting point. Perhaps I would of been fine sat in the lectures, but with nothing but the notes it was a little too much. In search of a more thorough resource, I stumbled across a wonderful find (which I now realise I unwittingly used in my study of complex numbers. MIT (yes the Massachusetts Institute of Technology) have issued an entire course. I just cannot praise this enough – it goes through all the basics and takes you through each topic in small, bitesize chunks. As a part timer this is wonderful.

You can think of your life as a jar full of marbles. The marbles are the full time job. You have a choice; you content yourself with the fullness of your jar, or you fill the gaps. Add sand until your jar is full. And then when you are done, fill the space with liquid…. your jar is starting to look a little fuller. This course provides the sand and the water. Lots and lots of nuggets of information you can fit around a busy schedule. What makes this course special is the problems – reems upon reems of problems. There are small check your understanding style problems, right up to an exam at the end of each section. With a little discipline I found this course was very very close to being back at univerisity (but considerably cheaper).

Overall, I have been a little spoilt this post – and every post won’t be this luxury. This course is perfect for anyone looking to gain some skill with DEs to a university level, and comes highly recommended from me.

taschimowitz@rationalisingtheuniverse.org

11 responses to “Ode to ODEs”

      • Hmmm…”getting” better? Again, hmmm…does that even work with non-linear time???😄😄

        I am fascinated by 99.9% of your posts…they are, each one, so beautifully and simply stated that even an old fossil like me (equates with pre-school level!) can understand most of what you’re saying. Oh! And I had read the post on time travel on Mekhi Dhesi’s blog (and just read it again!). I’ve spent several hours going back and forth on her posts, ever since you suggested/referenced her.

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      • Yes it is one of the best summaries that I have come across; a subject that must surly fascinate any mind. I am very glad that you enjoy them! I was worried when I stared this I would be unable to get the thoughts from my brain out onto the screen

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  2. Was mathematics invented or discovered ? Reading what the experts say it seems they are surprised to find such an obtuse subject should fit there investigations into the natural world. Branches of mathematics that seemed to be useless when first discovered were found to fit natural events. Complex numbers unlocked alternating current theory. Chance and probability theory aided the understanding of particle physics.
    It seemed as if nature herself was mathematical and waiting for mathematicians to unlock her secrets.
    But nature was more than mathematics for the investigative human mind ; she was beauty , poetry, religion, mystery . So powerful was this that some questioned that the human mind could have just come into being by the forces of the natural world.
    ‘ A thing of beauty is a joy forever ‘

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    • Mathematics is simply a description of nature and a very elegant and powerful one at that. It really is a fascinating study and one which is wholly rewarding. I think people often think mathematics is invented because humans do invent new ways of doing mathematics; but rather than it being invented it is thinking of new ways to express that which is already there

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  3. What about that which had not yet been discovered to be there, but when it is discovered mathematics appears to fit it’s basic nature.
    Conversely some natural phenomenon which we have not the mathematics to explain.
    Am I correct to say pure mathematics is simply mathematics that is not yet applied? or is there a pure mathematical realm which bears no relation to anything and exists in the mathematicians mind alone?

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    • I think it is correct to say that very little is genuinely pure mathematics in the most literal sense because it always has an application somewhere or another. But the split is along the lines of applied being more like modelling and predicting and things like proving number theorems in the realm of pure. Things like Topology as we have seen in the recent Nobel prize run can very quickly become very applicable; so you raise a very interesting point

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  4. We have Fermats last theorem it seems pretty pure to me. I could ask why do such a simple sequence of natural numbers prove so fascinating. Where do these extraordinary properties all come from ? or is there nothing extraordinary about them?Are the natural numbers a creation of the Mind of man? or is there something rooted in reality itself that mathematicians should hold them in awe?
    The Queen of mathematics where does this monarch reign? in nature or in man ?

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    • Natural numbers are most certainly not the creation of man; they actually are so called natural because they correspond to the discrete numbering systems we can assign to every day objects. Even animals have been known to recognise natural numbers. Nature will always be the queen, we seek to understand the best we can

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