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Can we say that Differential Equations is Calculus 4 in disguise?
The definition of the derivative involves the operation of taking a limit, and sometimes the limit problem is a derivative in disguise.
The most basic differential equations are the ones which you can just integrate to get the answer.
A typical Calculus textbook has, in the back so would be covered in Calculus III, has a section on separable first order differential equations, but I would NOT consider differential equations as part of Calculus itself since it covers many things that may use Calculus but are not part of Calculus itself. I would hope that, in addition to Calculus, Linear Algebra was a prerequisite for differential equations because the set of all solutions to a "homogeneous linear differential equation" forms a vector space.
Conceptually, calculus can include a lot more than can be discussed in two or three courses. Those introductory courses are general in nature and cover limits, derivatives, integrals, and series as well as examples and theorems relating to those concepts.
Many calculus courses do discuss a few of differential equations such as y′=ky and y′′=−y, but they don’t have time to do much. The concept is introduced, some applications, and the technique of separation of variables. That’s really no more than the first week in a course on differential equations.
From this I gather that differential equations is not easy at all. I know that knowledge of integration is needed to succeed in an ordinary differential equations course.
Question:
What is the basic difference between ordinary differential equations and partial differential equations?