Question

Do all differential equations contain derivatives or can they also contain other linear operators instead of the derivative operator?

Answer 1

Officially, a differential equation is a relation between the unknown function and its derivatives.

Answer 2

If it has no derivatives, its a normal equality :)

Answer 3

I think im confused by the question

Answer 4

In addition to the differential operator, they can of course have other operators.

Answer 5

Yes, differential equations can contain other linear operators in addition to the derivative operator.

Answer 6

wow im bored

Answer 7

Using non-linear operators would result in a non-linear differential equation :)

Answer 8

Using partial differential operators would result in a partial differential equation :)

Answer 9

Whenever your function is a function of multiple variables, indeed :) Though, I guess using a partial differential operator in other cases would be a bit purposeless.

Answer 10

I think I just reduced the average iq in this room lol. I don't wanna derail this question, but I'm doing calculus this semester. Should I bother trying to memorize all the trig identities I can find or should I just keep a page of them handy?

Answer 11

Remember a few important ones and you can deduce the rest in general :) Also, have a good look at the unit circle.

Answer 12

Thanks :).

Answer 13

So for instance a rotation by a matrix is a linear operator. Suppose I decide to have a "differential equation" where some of the terms are first derivatives, second derivatives, rotations by 30 degrees, rotation by 60 degrees, and just a multiple of the equation.

What's that crap called if anything.

Answer 14

a system of differential equations?

Answer 15

Rotation only means something in a multi-dimensional setting. In general, the amount of equations you get is equal to the amount of dimensions (assuming you are looking at one body as a whole).
The order of a differential equation is given by the highest order derivative, so if you have second order derivatives, you will have a second order differential equation.
If you use transformation matrices you will have to look into tensor calculus for the exact interpretation, but it will mean that you get derivatives with respect to the (x,y) components in 2d, or (x,y,z) in 3d. As your function is a function of multiple variables (your coordinates), it is called a partial differential equation.
You will have a second order partial differential equation, which, when written out, consists of 2 (2d) or 3 (3d) equations. I do not see any non-linear operators, so you could call the 2nd order PDE a linear one.