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I have already taken an ODE course and I passed it well but I learned nothing after passing it. I mean I can solve problems by applying the techniques I have been taught on them, but there are still a lot of things I don't understand such as:

  1. Why does Fourier series gain so much attention when it's possible to write other orthonormal expansions as well?

  2. I like to learn more about Laplace transformations and other types of transformations that my course did not cover (such as Z-transformation, for example)

  3. I like to learn more about special functions that arise in physical problems.

  4. I like to learn about famous problems in physics, mathematics and engineering that opened the door for new insights in mathematics.

  5. I like to learn about PDEs without going into too much detail like functional analysis, etc.

  6. Does every ODE/PDE come with a natural choice of orthonormal expansion? Or what is the relation between a differential equation and expressing the solution as a series? (Do I make sense here? Because I suspect that there must be some sort of relation, but I can't formulate it or find it on my own)

  7. Links between ODEs/PDEs and complex analysis

In short, I'm looking for a book that emphasizes important concepts behind differential equations and has a lot of good problems with applications to physics and engineering too. If the book comes with a manual, then that would be perfect. Any suggestions?

Edit: It doesn't have to be just one book. If you can suggest two books that complement each other and answer my question, that's fine.

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    $\begingroup$ You might take a look at Strang's book Differential Equations and Linear Algebra. $\endgroup$ – littleO Nov 11 '19 at 0:18
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I really enjoyed Differential Equations with Applications and Historical Notes by George Simmons. I think it’s a fantastic introduction to ODEs and it really attempts to give you a clear picture of the rigor that you need to do mathematics. It also introduces a lot of Physics problems as a part of the applications, so I think you’ll enjoy that as well.

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i think Tenenbaum and Pollar's book Ordinary Differential Equations would suffice to a certain degree for it covers:

  1. Application problems( mainly physical and geometrical ones, but also some other related to engineering)
  2. Examples and solutions to problems
  3. proofs of the devises and procedures used to solve Differential equations
  4. Gamma Function, legendre and laguerre equations
  5. Laplace Transform
  6. Numerical methods

unfortunatelly, it lacks the Z and Fourier transformations and the linear algebra approach you wanted.

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