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I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify.

By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc.

By logically coherent, I do mean that the writer goes beyond the "scratchwork" (Phase 1) and does the remainder of the problem (Phases 2,3 and 4).

For example, here's a more or less acceptable solution to the problem $y' = y.$

Phase 1 - Scratchwork.

Assume $y' = y$. (Scratchwork always begins with the assumption of the equation to be solved).

Assume also that $y \neq 0$. (In the scratchwork phase, you can just assume things like this without justification).

Then $\dfrac{y'}{y} = 1$, or in other words $\dfrac{1}{y}\dfrac{dy}{dx}=1$. Therefore, there exists $C$ such that $$\int \frac{dy}{y} = x + C.$$

Thus, there exists $C$ such that $$\log y = x + C.$$

This same $C$ must therefore satisfy $y = e^x e^C$.

Thus, there exists $C$ such that $y = Ce^x$.

Conclusion: For all real $C$, we have a prospective solution of the form $y = Ce^x$.

Phase 2 - Soundness.

We will show that for all real $C$, if $y=Ce^x$, then $y'=y$.

Proof. Assume $C$ is real and that $y=Ce^x$. Then since $y = Ce^x$, it follows that $y' = Ce^x$, thus $y'=y$, as required.

Phase 3 - Proliferation.

This is a phase that is sometimes needed, wherein we produce new solutions from the one's we've already found. e.g. if we only knew that $y=e^x$ was a solution, then we could use the linearity of the DE to show that $y=Ae^x$ is a solution. This isn't necessary, in this particular case.

Phase 4 - Completeness.

We will show that for all real $C$, if its not the case that $y=Ce^x$ everywhere, then its not the case that $y'=y$ everywhere.

Proof. By [Insert Theorem Here], the result follows.

goblin GONE
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    I hope this isn't utopia. I'd like a book like this, unfortunately people who work on that don't seem to enjoy being clear. – Git Gud Mar 10 '13 at 12:10
  • Note that what your scratchwork actually says is (provably) that any solution to $y' = y$ satisfying $y \neq 0$ also satisfies $y = C e^x$ (for some $C$). Or equivalently, that all solutions to $y'=y$ either satisfy $y = C e^x$ or are somewhere zero (or both). –  Mar 10 '13 at 12:29
  • @Hurkyl Yes that's true, but is it useful? The purpose of the scratchwork is to motivate a theorem; the fact that it gives us a wimpier version of the completeness result isn't the critical fact. – goblin GONE Mar 11 '13 at 08:35
  • The book "Applied Differential Equations" by Spiegel is one I liked. – Pedro Apr 17 '13 at 20:14
  • @PeterTamaroff, is it very "logical"? Carefully distinguishing between "If $A$, then $B$," versus "If $B$, then $A$," for example? – goblin GONE Apr 17 '13 at 22:57
  • Yes, sure. ${}{}{}{}$ – Pedro Apr 17 '13 at 22:59

1 Answers1

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I started studying ODE for the first time about a week ago and today I stumbled upon this book called An Introduction to Ordinary Differential Equations by Earl A. Coddington, at my school's library. Judging by the reviews on the link, apparently this is a classic.

I think it is exactly what you (we) want. It doesn't look like an analyst wrote it at all. Before I checked I was sure the author must have been an algebraist.

From the little time I spent with it I reckoned the book is very clear, the proofs are complete and detailed. Every random page I opened seemed to make sense. I'm definitely buying it.

The downside is it only talks about ODE.

I even like differential equations now!

Git Gud
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  • I second the nomination. The earlier book by Coddington and Levinson is considered a classic by specialists. But this one written later by Coddington alone is more appropriate for those not intending to use ODEs for research. – GEdgar Apr 17 '13 at 20:33
  • @GitGud, awesome! I'll definitely check it out in the next couple of months. Hopefully your first impressions of the book are accurate... – goblin GONE Apr 17 '13 at 21:56
  • This is indeed a good recommendation! +1 – amWhy Apr 18 '13 at 17:16