# What exactly is the relationship between calculus and differential equ

When I look up curriculums at school, I always find the main sequence of Calc 1-3(single variable to multivariable) then differential equations. But when I look up the definitions, diff equations are just called "equations and how they relate to their derivatives." I'm assuming there's more to it than that. What makes diff eq have its own course rather than just putting it in the main calculus sequence?

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AlephZero
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The course names are just something that has emerged as a sort of "standard" in the USA. When I learned this stuff (quite a while ago, in the UK) nobody talked about separate courses called Pre-calc, Calc 1, Calc 2, etc. In the UK, all the math courses up to high school level are just called "maths" (and the UK doesn't call high schools "high schools" either!)

Differential equations can involve derivatives of a single variable, or partial derivatives of several variables (usually called ordinary differential equations or ODEs, and partial differential equations or PDEs). Calc 1 and 2 are prerequisites for ODEs and Calc 3 (multi-variable) for PDEs. So they are "part of the main calculus sequence", except the course isn't called "Calc 4".

There are two sorts of diff eq courses: those that just teach you how to solve them, and those that also teach you some mathematics. The more mathematical courses need linear algebra as a prerequisite as well as calculus.

Adding to AlephZero's response, most calculus sequences do teach the basics of solving differential equations. Normal integration can be thought of as solving the simplest differential equations. But, it gets very complicated quickly and finding solutions to differential equations are very important to most sciences, so an entire semester is devoted to at least ODE's.

I have a friend that went to a school where they had Calc I-IV instead of calling Calc IV a differential equations class. Both Calc III and IV required linear algebra. I forget which school it was, but the course was the same except that it took longer to get her transfer approved I see, thanks for the responses