Rigorous Differential Equations text

In summary, the conversation discusses the search for a rigorous differential equations textbook with a focus on proofs and theoretical questions. Suggestions for books include Ross's "Differential Equations" and Teschl's "Ordinary Differential Equations," with Arnol'd's "Ordinary Differential Equations" also mentioned as a good choice for exploring connections between differential equations and geometry. The conversation also mentions the importance of understanding introductory material before delving into more advanced works.
  • #1
SrVishi
75
15
Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would like theoretical questions if possible too, and applied as well (I guess it would be hard to find a differential equations text without applications haha). Could anyone tell me if there are any such books? If not, could you at least tell me what your favorite is? I hear Differential Equations with historical notes by Simmons is a good one. Thanks in advance for any response. Sorry if this question is worded weirdly. I'm on mobile in public and can't focus well.
 
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  • #2
What do you know of differential equations already?
 
  • #3
I already had an introductory course in Ordinary Differential Equations, though it was kind of a crash course, so I would kind of like to review it, with a more rigorous flavor of course.
 
  • #4
Well, you probably can't do better than Ross's differential equations: https://www.amazon.com/dp/0471032948/?tag=pfamazon01-20 This is one of my favorite books out there. Be sure to get the 800 page version, since it contains more theoretical stuff.

Now, the book is very good, and contains quite a lot of theory, with of course the existence and uniqueness theorems. But it isn't really comparable to Rudin in terms of sophistication. But I feel that one should understand the material in this book before getting to a more advanced work, since that advanced work will likely take a lot of things for granted that are in such "introductory books".

For a more rigorous and advanced book, I recommend Teschl: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ which is freely available on his website, but you can buy it too: https://www.amazon.com/dp/0821883283/?tag=pfamazon01-20 This book will not care about solving differential equations though.
 
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  • #6
You can start with Coddington ( introduction), I am familiar with this one. Then you can try his differential equation book(have not looked at it yet, my understanding is not there yet).
 

Related to Rigorous Differential Equations text

1. What are differential equations and why are they important in science?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are important in science because they allow us to model and understand the behavior of complex systems, such as those found in physics, engineering, and biology.

2. How does one solve a rigorous differential equation?

Solving a rigorous differential equation involves finding the function that satisfies the equation and any given initial conditions. This can be done using various mathematical techniques, such as separation of variables, substitution, or using numerical methods.

3. What are the applications of rigorous differential equations?

Rigorous differential equations have numerous applications in science, including modeling population dynamics, predicting the motion of celestial bodies, analyzing chemical reactions, and understanding electrical circuits. They are also used in engineering to design and optimize systems.

4. What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Additionally, partial differential equations often involve systems with more complex behavior and require more sophisticated methods for solving.

5. Are there any real-world limitations to using rigorous differential equations?

While rigorous differential equations are a powerful tool in modeling and understanding complex systems, they do have limitations. For example, they may not accurately capture the behavior of systems with chaotic or stochastic elements. Additionally, there may be numerical errors or uncertainties in the initial conditions that can affect the accuracy of the solutions.

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