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.