With calculus, you are already well into ordinary differential equations -- partial differential equations are different, but with calculus you have a good start on the basics of those, too.

A simple ordinary differential equation is below where of course just from calculus

y'(t) = d/dt y(t)

and the equation is

y'(t) = k y(t) ( b - y(t))

So, for the context: t is time, say, in seconds. y is some real valued function of t, that is, y(t), b and k are constants. We are given the value of y at 0, that is y(t) at t = 0, that is, y(0). We want the value of y(t) for t > 0.

Okay, with that, just need to use freshman calculus, for positive time s, integrate y'(t) from 0 to s. This is a simple exercise, without looking up the last dozen times I did that, maybe use integration by parts or some such. End up with a quotient with some exponentials.

Differential equations pop up in motion, e.g., from Newton's second law, AC circuit theory, and some other areas of science and engineering.

Boundary value problems, e.g., vibrating stings, parts of deterministic optimal control, are closely related but, still, significantly different.

Long some of the pure mathematicians went, in a word, "nuts" studying differential equations. The best of the results are good, and some of those are nicely useful. In places the work has nice contact with linear algebra, matrix theory, and vector spaces of functions, functional analysis, e.g., Hilbert and Banach spaces. But hanging over the whole subject is a suspicion that, really, as nice as the general theories are, mostly the applications are just a few, standard differential equations. It's a little like learning everything about civil engineering when really are only going to do framing carpentry, hang drywall, and apply roof shingles.

Once I bought

Garrett Birkhoff and Gian-Carlo Rota,
{\it Ordinary Differential Equations,\/}
Ginn and Company,
Boston,
1962.\ \

I looked through it, saw lots of intricate stuff, but wondered just why I should dig into that. Since then I read a story about Rota about how, apparently, he felt much the same about the material, got stuck teaching the differential equations course because he wrote that book, and wanted, essentially, to f'get about that book and its material!

I had a full college course in ordinary differential equations. Okay: It left me wildly over educated for the differential equations in AC circuit theory. Otherwise I didn't much like the book, the teacher, or the course.

On the advanced stuff, here is some more

Earl A.\ Coddington and
Norman Levinson,
{\it Theory of Ordinary Differential Equations,\/}
McGraw-Hill,
New York,
1955.\ \

It has a nice result of Caratheodory, but in general could lose a lot of sleep working through that!

I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT,

Francis B.\ Hildebrand,
{\it Advanced Calculus for Applications,\/}
Prentice-Hall,
Englewood Cliffs, NJ,
1962.\ \

So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material seriously I wanted to see the proofs. Now that I know a lot more math, no doubt some of it originally motivated by material in that book, maybe I could fill in the proofs.

When I was at FedEx, I wondered about the cheapest way to climb, cruise, and descend the airplanes, had heard about

Michael Athans and
Peter L.\ Falb,
{\it Optimal Control:\ \
An Introduction to the Theory and Its Applications,\/}
McGraw-Hill Book Company,
New York,
1966.\ \

and flew up to MIT and met with Athans, got his course notes, etc. He explained that an application would be a "two point boundary value problem with mixed end conditions" -- okay, I'd had a course on numerical methods for that. But, in the early parts of the book will see something interesting -- fast, and well written coverage of the differential equations material needed for the book. This is an example of a general situation: Sometimes the best place to learn something is in an introduction or appendix written by a real expert who is also a good writer, intended as background for the rest of the book. So, such a source cuts out the tangential, maybe curious cruft can't much hope to use.

At one point after college
on my own I carefully read, not nearly new at the time (TeX markup):

Earl A.\ Coddington,
{\it An Introduction to Ordinary
Differential Equations,\/}
Prentice-Hall,
Englewood Cliffs, NJ,
1961.

Coddington was not just a grand expert in the field but also a good writer. I really liked his stuff on variation of parameters -- a bit amazing. Note: Can find mention of that in the famous movie The Day the Earth Stood Still -- apparently that math was hot stuff in applied math about when the movie was made.

Can say some quite similar things about partial differential equations -- e.g., there are deep books, some connections with functional analysis (and even the theory of distributions) but the main interests are the partial differential equations of mathematical physics, especially, Maxwell's equations, the heat equation, the wave equation, Schrödinger equation, a wave equation, and the notorious Navier-Stokes equations -- which likely should attack only for limited goals and in somewhat special cases.

Net, unless you have some significant reason for more, I suggest you learn what you need to know, just in time, when and if you need it. But, in that case, as elsewhere, a good pure math background in calculus, and advanced calculus with the proofs, etc. will be good to have.