Take the initial value problem
$$y'=\cases{x\bigl(1+2\log|x|\bigr)\quad &$(x\ne0)$ \cr 0&$(x=0)$\cr}\ ,\qquad y(0)=0\ .$$
This example obviously fulfills the assumptions of the existence and uniqueness theorem, so there is exactly one solution. As is easily checked this solution is given by
$$y(x)=\cases{x^2\>\log|x|\quad&$(x\ne0)$\cr 0&$(x=0)$\cr}\ .$$
This function is not analytic in any neighborhood of $x=0$.
I think the best thing for you to do is to talk to your academic advisor if you haven't done so already.
The reason I say that is my guess that an actual professor will have more information on:
- the requirement and customs of this specific department
- general availability of the classes you mentioned and frequency with which they are offered
- the order in which students from your cohort usually take classes
- you personal preferences and proclivities
That is assuming you have an advisor who is relatively easily accessible and who will be willing to spend time on a brief conversation with you.
This is the best case scenario, as in addition to aforementioned advantages related to the access of information, talking to the advisor gives you an option for real-time dialogue with the person advising.
If, however, talking to academic advisor is not a viable option for you, I would recommend taking the courses you listed in the following order:
- real analysis
- complex analysis/differential equations
- numerical analysis
I assume you already have solid background in (multivariable) calculus and linear algebra.
If not, you should take these to ASAP, and definitely before you take anything from your list.
Also, I want to point out again that whatever I wrote should be taken with a grain of salt, as it is intended to be a "rule of thumb" list.
In order to make proper decision on which classes to take when, you should take into account a lot of individual-specific factors, included but not limited to
- you background
- academic interests
- customs of university and department
- who is teaching which class any given semester
- which topics are covered in each of the courses
- whether it is possible (and acceptable for you) to take several classes simultaneously
- $\cdots$
- etc.
Best Answer
In the same way that not all algebraic equations can be solved using algebra. For algebraic equations, you have Galois theory, and the solvability of an algebraic equation by algebraic means can be studies by studying the Galois group of that equation. Similarly, for differential equations one has differential Galois theory, and the deeply related Picard-Vessiot theory. Liouville's theorem also gives an answer that you might find useful.