Introduction to Differential Equations (2008)

I think the calculus of variations might be a better approach to introducing ODEs in first year.

You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation).

This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.

Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you've motivated ODEs and given the students are reason to solve ODEs with constant coefficients.

Not really a website, but I believe you can find ebooks online...

I highly recommend Numerical Linear Algebra by Trefethen. It gives very detailed descriptions of particular interpretations of the singular value decomposition and eigenvalues, it works out detailed algorithms for LU factorization, eigenvalue/eigenvector decomposition, QR factorization etc. If you know basic linear algebra, the book is a pleasure to read through. For this crowd of people it is also very practical.

I don't know a good resource for probability... it is a much more diverse subject than linear algebra (which is a very small, very detailed subset of algebra).

The course overview is here[0]. I haven't looked at how in-depth the later sections are, but I can't think of a topic related to differential equations that I used in my undergrad physics degree that isn't at least touched upon here.

[0]: http://tutorial.math.lamar.edu/Classes/DE/DE.aspx

Any type of graph you have, you can use it, just like integrals, to calculate areas etc.

For a lot of things in life we plot graphs. Your software might do it transparently to you, but how it works internally is using mathematical concepts. It is good to know it and I really enjoyed learning it at college, as it opens your mind about how things work. But I don't believe it is a must to know.

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks that they taught us to solve them.

10. TEACH CONCEPTS, NOT TRICKS

What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality, and the justifications I have heard of it, citing poor preparation of the students, their unwillingness to learn, and the possibility of assigning clever problem sets, are lazy ways out.

In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach.

We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations; such information can nowadays be gotten in much better ways than sitting in a classroom. A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course.

https://web.williams.edu/Mathematics/lg5/Rota.pdf