After taking Diffy Q I feel like my math intuition has gone way down

Main Question or Discussion Point

I took Diffy Q about a year ago and that was the first math class that felt like a cookbook class. Ever since that class I feel like I've gotten worse and thinking less intuitive about math. I think I got discouraged because so many of those methods for solving in Diffy Q just popped up out of nowhere from someone much smarter than others (Euler/Lagrange). Now when I see a differential equation I just look up the method to solve it and apply. I can analyze my solution afterwards and see if it's correct but isn't there a better way to see a solution without these formal methods? I looked into geometry and differential equations because I'm a highly visual learner but those books/topics look far advanced for what I need.

Now I'm learning about Laplace Transforms and we derived a couple easy ones but now we're gonna use a table to figure out the more complicated solutions.. My goal is not rigor but strong intuition in math and physics. I don't mind using a table for quick reference but I'd rather like to at least feel the correct solution from a intuitive geometrical sense. What can help me bring back my intuition of math?

This semester I'm taking a math class on perturbation theory, transforms, and special functions and my hopes are it will help my math thinking but I'm not sure it will yet. I think I learned more about calculus and intuition from my physics classes than math classes. Has anyone else ever felt this way?

In maths I feel that this is pretty normal. You start off with elementary calculus and elementary linear algebra that involves very litter rigour or complexity and you develop quite a good intuition of these things. Later on you delve into greater complexity, rigour and generality, and your intuition of these things drops down, only because these things are now 'generalised' - you can no longer use the intuition you had in the elementary cases with as much certainty. Even later you start building up even higher levels of abstractions and you slowly start to see these things you once thought were complicated and relatively arbitrary become simpler concepts which just generalise the elementary cases you studied earlier on, and your intuition starts to grow again.

In physics however there is rarely very much rigour given and physical arguments will usually suffice and for this reason there will be a good level of intuition at (mostly) all times.

Thats how I see things anyway - others may differ.

I don't see how you can have intuition for something that is abstract...?

Now when I see a differential equation I just look up the method to solve it and apply.
That's your mistake. I can derive most of the basic solutions just from understanding the principles involved. It's really not that difficult.

The math professor who taught me differential equations said that the most common method math professors use to integrate difficult integrals is to assume a solution that looks close to what the solution should be, then work backwards to fill in the blanks. In light of that mentality, it should be no wonder why the same method is applied to differential equations.

I can analyze my solution afterwards and see if it's correct but isn't there a better way to see a solution without these formal methods? I looked into geometry and differential equations because I'm a highly visual learner but those books/topics look far advanced for what I need.
Look into numerical methods for solving differential equations. Once you wrap your head around what's going on, the equations make perfect sense. Differential equations map points to vectors. The solutions to differential equations follow the lines formed by the vectors. Admittedly the second-order equations start to become hard to visualize, but...

differential equations is not a neat and tidy subject. there is no generally-applicable method for finding closed-form exact solutions in terms of familiar "elementary" functions, such as polynomials, sine, cosine, etc. the techniques that do exist are very ad hoc, and are always limited to particular types of equations.

that's just the nature of the subject. it makes a mess faster than you can clean it up. it is trivial to come up with a new equation that falls outside a particular solved family. but it is often difficult, if not impossible, to solve that new equation. indeed it is often demonstrably true that methods which work for one type of equation can't work for another.

Read Chapter 1 of Visual Complex Analysis to really understand complex numbers. That can help with second order linear equations because those use complex exponentials. Feynman has a good discussion of vibrating springs in Volume 1 of his lectures. If you put that all together, you should understand 2nd order linear equations pretty well in most cases.

Secondly, as Angry Citizen mentioned, to get the general idea of what diff eq is about, you can think of vector fields that tell you where to flow to.

Thirdly, you have to understand linear algebra well. Unfortunately (or maybe fortunately if you like lots of linear algebra), Jordan canonical forms show up and are the secret explanation of for those repeated roots solutions that Boyce and DiPrima would try to randomly shove down your throat with no motivation what so ever. They can be excused in this particular case because of the sophisticated linear algebra involved. Jordan forms are somewhat intuitive once you understand it well enough, but not directly visualizable. Understanding it involves some visualization for me, but only highly abstract visualization that can only be described once you know some linear algebra concepts well enough.

As for Laplace Transforms, when I was an EE major, they annoyed the hell out of me, since they were so unmotivated. But it wasn't so bad when I studied signal processing and learned about Fourier series, and Fourier transforms. I would basically think of it as a modified version of the Fourier transform, now, but I would like to find the original motivation for Laplace, which seems to have been something to do with probability or statistics.

Admittedly the second-order equations start to become hard to visualize, but...
Actually, it's not that hard if you have some higher-dimensional intuition. It's just a vector field on the space of based vectors (otherwise known as the tangent bundle). People mean different things by visualization. If you ask a mathematician if they can visualize 10-dimensions, the standard answer would be no, but if you phrased it as a question of "can you think about it visually", the answer would be yes, for sure, at least for some aspects of things. Also, as I mentioned, you can get a lot of intuition about 2nd order equations from abandoning the vector-field picture and thinking about springs.

I don't see how you can have intuition for something that is abstract...?
And I don't see why not. Math is very abstract and intuition is a big part of it. Probably, the only reason you think you can't is because people don't care about motivation, so they just stupidly throw definitions at you. To me, throwing a definition at someone without motivation is sort of like giving a theorem without proof (which is a valid option sometimes, but it's obvious to everyone that something is being left out, whereas with unmotivated definitions, people act like nothing has been left out, and that definitions, unlike theorems, come down to us from the sky, which is just absurd beyond belief). Not to mention unmotivated proofs, which seem completely non-intuitive and abstract because there is no hint as to how you would come up with them, but often, the proofs become obvious if the idea behind them, rather than the formal logic, is explained.