Differential Equations

• Course notes are here or here (both unofficial)
  • A guy on YouTube made lectures for the course based on the notes
• Example sheets and practice for integrals.

• J. Robinson An introduction to Differential Equations. Cambridge University Press, 2004
• W.E. Boyce and R.C. DiPrima Elementary Differential Equations and Boundary-Value Problems. Wiley, 2004
• G.F.Simmons Differential Equations (with applications and historical notes). McGraw-Hill 1991
• D.G. Zill and M.R. Cullen Differential Equations with Boundary Value Problems. Brooks/Cole 2001

We need to learn Peter Lax's Calculus with Applications which introduces differential equations.

Other options:

• This course has recorded lectures
• A problem bank with solutions
• The SciML book and YouTube lectures to fill in the discrete differential equations content
  • Teaches Julia for the computational projects we'll have to do later too.

I'll do them too.

Go to one of the shadow libraries and find Anna's archive. Find Calculus With Applications second edition by Peter Lax, Terrell, Shea. Skip all the most likely criminal links trying to get you to pay for this pirate download and select a 'slow link'.

You can buy a copy of the second edition on Abe Book for $35 and I recommend doing so in order to start creating a library because shit just disappears these days and nobody can remotely ninja edit or delete your physical copy.

The reason I chose this book above all others is the simplicity of the definitions. Today in every calc text you will be forced to read about silly abstract nonsense like Dedekind cuts (Axler) or limits of Cauchy sequences (Tao, all other authors) which are an annoying distraction from the applied calculus of differential equations which Lax was an expert. Instead we will learn the 'infinite decimal' version of real numbers which makes all the proofs laughably simple because you are only comparing decimals and declaring them equal after you have decided the precision is good enough. Real numbers don't exist in computers so imagine these are simply truncated approximations and we will learn much later all the so-called modern definitions of reals in graduate analysis courses. Real numbers themselves are totally bullshit anyway so who cares. A good way to think about them is imagine things like the square root of 2 are algorithms that are attached to the rational number line and these algorithms never terminate. The same goes for Pi and the irrational constant e which cannot be represented as a ratio either but we can easily generate e with a series to any decimal point needed for precision.

The strategy here is learn chapter 1 through 9 until we get to differential equations then we can resume the regular Cambridge tripos.

Peter Lax once convinced the Courant Institute at NYU in 1969 to purchase a supercomputer (today worth $30m) to find approximations to partial differential equations (now we can do this with AI) and communist retards took it hostage demanding a ransom payment. They wired the computer with IEDs in 1970 which Lax being one of the world's greatest physicists disarmed himself at great risk. He married the daughter of Richard Courant.

We have to learn the Reimann constructive integral which was abandoned by pure mathematicians 100+ years ago. Wait what? Yes. It was replaced by the Lebesgue integral. In the 1960s two mathematicians Ralph Henstock and Jaroslav Kurzweil both independently invented the 'guage' integral a generalization of the Lesbesgue integral that doesn't need all the apparatus of measure theory which we'll learn here too. This is now called (2007) the Natural Integral.

TODO