TITLE: What is the application/use to know about the order and degree of a differential equation?
QUESTION [5 upvotes]: A degree of a polynomial tells us the maximum number of possible roots of the equation.
But what is the use to know about the order and degree of the differential equation. Like, what does it really tell us about the equation we are working with?
For Example: If I am working with a polynomial of degree 3, and I tell that to you, you will be expecting, at most 3 roots of that polynomial. Degree of a polynomial only tells us about the number of roots, and so, x3 and x3 + 3x2 + 5 both are equation of degree 3 and tells it will have a maximum of 3 roots.
But what does telling you that I am working with a differential equation of order 3 and degree 2 really tell you about the equation because you can not predict what the equation is, but then what can you predict with the data I give you?
I am new in the Journey of Differential Equations and wanted to know about the things (and reasons) that I am learning.
Thanks in advance for your answer. :)

REPLY [7 votes]: The order is a fundamental piece of information for a scalar ODE that tells you the dimension of the state vectors in the equivalent first order system. (For a system of higher order ODE you would want the sum of orders.)
The degree is a much more controversial idea, I'll refer to earlier answers of mine on this topic

What is the definition of the degree of a differential equation?
Degree of a differential equation