James2 said:
Alright, I know my algebra, geometry, and plane trigonometry. I've been calculating derivatives for a while now, but I wish to get to a higher level of differentiation. Say, differential equations. Could somebody explain how to solve a differential equation?
Hey James2 and welcome to the forums.
This question that is seemingly so simple, is actually not an easy one to answer.
The thing is that not all differential equations were created equal.
In some cases you can use some rather simple substitutions or calculus identities (like the chain rule) to get the answer and for others, you have to resort to things like integral transforms like the Laplace transform.
Even then with the mathematical machinery that we have, many differential equations like non-linear ones just can't be solved with the techniques we have at the moment and because of this we need to not only use computers, but we have to understand the theory that makes getting a solution actually work to begin with.
So keep the above in mind: in other words, we can't always solve a DE like we solve say a set of equations like x + 2y = 4 and x + 3y = 7 where we get x = blah and y = blah: it's not that easy when you start getting to the complicated kind of equations.
In terms of understanding differentiation though, once you understand all the basics like the chain rule, product rule, quotient rule and others including partial derivatives, then the rest is pretty much using these rules in a given context.
