My textbook has this definition for an ODE:
If a differential equation contains only ordinary de-
rivatives of one or more unknown functions with respect to a single independent
variable, it is said to be an ordinary differential equation (ODE).
I cannot for the life of me find a clear definition for the construct "ordinary derivative". I assume this is important because it is one of two qualifiers in the statement which is defining the ODE. I cant be the only person to have ever searched this and yet my google foo is failing me. Super grateful for any help here, explanation or reference. A reference would be fantastic since i suspect this may just be an issue of hand waving in the definition and perhaps i should get a more rigorous textbook but i dont know what to buy. Thanks
Best Answer
"Ordinary" just means "not partial" (as in "partial derivative"), and in this context is wholly redundant with the phrase "with respect to a single independent variable". In other words, our function is a function of one (real) variable, and we can differentiate it with respect to that variable any number of times and write an equation relating those derivatives and the original function. This is in contrast with a partial differential equation, where we have a function of more than one real variable and can have an equation relating its partial derivatives with respect to the different variables.