Reddit reviews Differential Equations and Their Applications: An Introduction to Applied Mathematics (Texts in Applied Mathematics) (v. 11)

These were the most enlightening for me on their subjects:

• Differential and Integral Calculus, Vol. 1&2 by Richard Courant
  
• Linear Algebra by Georgi Shilov
  
• Differential Equations and Their Applications by Martin Braun
  
• Introduction to Geometry by H.S.M. Coxeter
  
• A Course in Mathematical Analysis, Vol. 1&2 by Edouard Goursat
  
• Introduction to Topology and Modern Analysis by George Simmons
  
• The Theory of Functions of a Complex Variable by A.I. Markushevich
  
• Basic Topology by M.A. Armstrong
  
• Topology by James Dugundji
  
• Ordinary Differential Equations by V.I. Arnold
  
• Partial Differential Equations by Fritz John
  
• Introduction to Probability Theory by Paul Hoel, Sidney Port and Charles Stone
  
• Linear Programming by Katta Murty
  
• Nonlinear Programming: Theory and Algorithms by M. Bazaraa, H. Sherali and C. Shetty
  
• A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz
  
• A First Course in Stochastic Processes by Samuel Karlin and Howard Taylor
  
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
  
• Homology Theory by James Vick

At the elementary level, Braun is good.

At the intermediate level, Arnold is the best.

These are my personal favourites for introductory books on ODEs - [Simmons & Krantz's Differential equations: theory, techniques and practice](https://www.amazon.com/Differential-Equations-Steven-Krantz-Simmons/dp/0070616094) is a great book with examples from physics and engineering along with lots of historic notes.

[Braun's differential equations and their applications](https://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/ref=sr\_1\_1?crid=35EOUTZZ32HDA&keywords=braun+differential+equations&qid=1556968795&s=books&sprefix=braun+Differenz%2Cstripbooks-intl-ship%2C215&sr=1-1) is another applications oriented differential equations book that is a bit more involved than Simmon's but has a much broader perspective with introductions to bifurcation theory and applications in mathematical biology.

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If you're not planning to do research in ODE theory, but want to learn the basic theory more rigorously, then [Hurewicz's Lectures](https://www.amazon.com/Lectures-Ordinary-Differential-Equations-Hurewicz/dp/1258814889/ref=sr\_1\_1?crid=341Z3D48AUTBU&keywords=hurewicz+differential&qid=1556969136&s=gateway&sprefix=hurewicz+%2Cstripbooks-intl-ship%2C216&sr=8-1) is a perfect short book that covers the basic theorems for existence and uniqueness of solutions of ODEs.

If you'd like some alternatives, I personally like Braun and I heard Arnold is good.

I'm in engineering, and back when I took DE I mainly used Braun for applications I think. I remember liking it quite a bit. http://www.amazon.com/Differential-Equations-Their-Applications-Introduction/dp/0387978941/

Edit: also IIRC, Churchill (Complex Variables and Applications) had a section about applications of DE in Laplace transforms.

Even in it's linear form the second order ODE is extremely general, there's just a ton of applications (and unfortunately the first one that came to mind was air drag, which has a (y')^2 term).

If you've done linear systems yet you'll know that this can actually be converted into a system of ODEs, if we allow y_1 = y and y_2 = y' we can see that this is then the vector equation

[y_1, y_2]' = [y_2, B/A y_2 + C/A y_1] + [0, f(t)/A]

So this actually tells us that we can model a large number of systems where there is some sort of mixing between the variables y and y'.

There is a rather cool application I found in one of the textbooks I have lying around though that is derived using second order ODEs (rather than the system approach).

Apparently the second order ODEs give a good model for the detection of diabetes. This is covered in chapter 2.7 of Braun's book, it is based off of chapter 4 of 1969 book by E. Ackerman et al. called "Concepts and models of Biomathematics".

Here is a sample paper that looks like the mentioned model.

Here is (what appears like) a more developed model using a 3rd order system.

> don't think that there is a logical progression to approaching mathematics

Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.

> Go to the mathematics section of a library, yank any book off the shelf, and go to town.

I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.

My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941

That is more than a summer's worth of work.

Sorry, agelobear, to be such a contrarian.

LIST OF APPLICATIONS IN MY DIFF EQ PLAYLIST
Have you seen the first video in my series on differential equations?

I'm still working on the playlist, but the first video lists a bunch of applications that you might not have seen before. My goal was to provide a sample of the diversity of applications outside of mathematics, and I chose fairly concrete examples that include applications in engineering.

I don't go into any depth at all regarding any of the particular applications (it's just a short introductory video), but you might find the brief introduction to be helpful.

If you find any one of the applications interesting, then a Google search will reveal more detailed resources.

A COUPLE OF FREE OR INEXPENSIVE BOOKS
Also, off the top of my head, the books below have quite a few applications that you might not see in the more standard textbooks.

• Differential Equations and Their Applications: An Introduction to Applied Mathematics, Martin Braun (Amazon, PDF)
  
• Ordinary Differential Equations, Morris Tenenbaum and Harry Pollard (Amazon)
  
  I think you can find other legal PDFs of Braun's third edition, too. Pollard and Tenenbaum is an inexpensive paperback from Dover, and I actually found a copy at my local library.
  
  ENGINEERING BOOKS
  Of course, the books I listed are strictly devoted to differential equations, but you can find other applications if you look for books in engineering. For example, I used differential equations in a course on signals and systems that I tutored last semester (applications included electrical circuits and mass-spring-damper systems).
  
  NEAT VIDEO (SOFT BODY MODELING)
  By the way, here's a cool video of various soft body simulations based on mass-spring-damper systems modeled by differential equations.
  
  Here's a Wikipedia article on soft body dynamics. This belongs to the field of computer graphics, so I'm not sure if you're interested, but mass-spring-damper systems come up a fair amount in engineering courses, and this is an application of those ideas that might open your mind a bit to other possible applications.
  
  Edit: typo

This is the best one I have used.