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I am looking to expand my toolkit for solving and understanding differential equations.

I often come across results and methods such as the possibility of solution by quadrature of ODEs, transformation properties of certain equations or Equivalence, Invariants, and Symmetry by Olver. My formal training in math has been an undergratuate degree in physics, so I am lacking in the rigorous math training needed to understand many of them.

Olver's text seems to be what I want to be learning: Lie Groups, symmetries of differential equations, differential invariants. In the preface to Olver's text he states that

The basic prerequisites for the book are multi-variable calculus specifically the implicit and inverse function theorems and the divergence theorem - basic tensor and exterior algebra, and a smattering of group theory. Results from elementary linear algebra and complex analysis, and basic existence theorems for ordinary differential equations are used without comment.

Are these the prerequisites I am after? I've come across questions such as these, though my aim is not necessarily to learn Lie Groups/differential geometry, it is to learn whatever advanced subjects aid in solving DEs. I would appreciate a knowledgeable voice on where I should/want to head.

Eli Bartlett
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  • If it’s your toolkit you’re looking to expand, Ordinary Differential Equations and Their Solutions by Murphy is a good start. – A rural reader Feb 01 '23 at 00:31
  • @Aruralreader a good source, though from a quick skim I am already familiar with and use the methods outlined in the book as I have been through the Handbook of Exact Solutions for Ordinary Differential Equations by Polyanin and Zaitsev a good few times – Eli Bartlett Feb 01 '23 at 00:42
  • Perhaps expanding my toolkit is the wrong phrasing. I'd like to be able to derive the results and come up with new ones such as that in the paper "Transformation properties of $\ddot x+f_1(t)\dot x+f_x(t)x+f_3(t)x^n$" – Eli Bartlett Feb 01 '23 at 00:56
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    I'd prefer Olver's older book, "Applications of Lie Groups of Differential Equations" for this as it is more oriented towards computing symmetries and using them to get solutions where as the book you're using is more oriented towards Cartan's Equivalence method. It's useful, but more fruitful for PDE. Whereas it's harder to use for ODE since many ODE do not have local invariants. Unfortunately, he doesn't cover some of those PDE methods in his purple book.

    I recommend consider Bluman's book: "Symmetry and Integration methods for differential equations" for this as well.

     – Doge Chan Feb 01 '23 at 01:25
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    Separate from Symmetry methods, there are the typical dynamical methods. I'd look into these too. For example, "Nonlinear dynamics and Chaos" by Steven Strogatz though this book doesn't quite clearly define things at times, "Differential Equations and Dynamical Systems" by Lawrence Perko, and "Mathematics for Dynamical modeling" by Edward Beltrami which is more oriented towards mathematical modeling with dynamics. You don't actually solve the equations in dynamics though, you simply just look for numbers that let you argue the local behavior of solutions without finding solutions. – Doge Chan Feb 01 '23 at 01:41
  • Thanks @DogeChan, you’ve spared me a toolkit more suited to PDEs, which are, technically speaking, less fun to solve haha. – Eli Bartlett Feb 01 '23 at 04:10
  • @DogeChan do you have any advice about how to learn the prerequisites for Olver’s text? – Eli Bartlett Feb 01 '23 at 04:11
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    For the general group theory side, I'd look for an elementary book like Judson's book "Abstract Algebra: Theory and Applications" which is free. For Lie group theory, it usually comes down to Lie algebras so something like "Introduction to Lie Algebras" by Erdmann and Wildon would help with some topics. For differential geometry, I'd try "Differential Geometry for Physicists" by Isham and "An introduction to Differentiable Manifolds and Riemannian Geometry" by Boothby. Lie Groups as manifolds are discussed in both books to a degree. – Doge Chan Feb 01 '23 at 04:39

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