i have a few questions to ask about differential equations ...
how many types of differential equations are there ... ?

sometimes i like to make up themes for my studies ...
few funny things went through my head ...when i saw this thread ,How is it that mathematics describe reality so well?

i also happened to read this ...

i was also wondering that if you learn enough differential equation , you might be able to understand things like " The Schrödinger equation " , the nature of reality ... and in the end how mathematics can describe reality so well ....

i am also looking for some advice on how to start learning differential equation properly ...??

No. The image you posted had nothing to do with differential equations or Schrodinger's equation, and that's why I deleted it.

Images that are included with posts can be helpful, but most of the images you have posted are not helpful, and just clutter it up. Apparently you get a bunch of images on your computer screen, and then take a screen shot of them all. The one you posted in this thread had two of the same images in it, neither of which had anything to do with the question you're asking.

Start with calculus and linear algebra, then continue with a book on ordinary differential equations like Boyce and diPrima. Avoid stuff you find on internet like handouts or movies, read a real book and make all the exercises.

I would also give an advice. It is important to understand that differential equations do not have a habit to be integrated explicitly. So do not think that the center of this science is a skill in integrating equations. It is important to be able to integrate standard types of integrable differential equations but the main thing is the qualitative analysis of DE

To confidently solve differential equations, you need to understand how the equations are classified by order, how to distinguish between linear, separable, and exact equations, and how to identify homogenous and nonhomogeneous differential equations. Learn the method of undetermined coefficients to work out nonhomogeneous differential equations

Boyce De Prima is a run of the mill book. I do however like Morris Terrebaun. I prefer Ross: Differential Equations. Crystal clear explanation and proofs are given. Make sure to purchase the book titled Differential Equations and not Introduction to Ordinary Differential Equations. Differential Equations is the complete book, which includes some nice theory chapter.

We characterize them into types like we do with integrals but the list is ultimately endless.

Mathematics is a creation of the human brain; the earth may have revolved around the sun before mankind but the mathematical constructs of gravity, in my humble opinion, did not. And of course the human brain is a product of evolution: survival and reproductive success of the organism in the environment. Organisms adapt to their environment: when in New York, act like a New Yorker. So too with the human brain. The hominid brain evolved as a survival strategy in the world early humans found themselves in. We cannot separate the human brain from the world it evolved in. The human brain and the world are cut from the same tapestry! Why should it be such a mystery the mathematics which emerges from our minds so well conforms to the reality of the world which created that mind?

I'll tell you my own personal experience with the matter: a long time ago I use to look outside my window and wonder why about a lot of things. About 20 years ago I started studying non-linear differential equations. I no longer wonder why about a lot of things. Not saying I know, rather, I'm simply saying I no longer wonder why. :)