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

Solution 1:

The "degree" of a differential equation is, in an introductory level course, a teaching aid to be able to formulate a greater diversity of classification exercises. Which is useful when learning the "language" of differential equations, to get an intuition on what is syntactically correct and what is symbolic garbage. As there is no theoretical follow-up there is also no sharp definition, what is expected as answer depends on the textbook or course material, so all what you cited is likely correct in the context it was used.

In an advanced, specialized context the idea of an degree also makes sense in an algebraic treatment of purely algebraic differential equations (which means no sine permitted). It tells you how many (complex-valued) solution curves pass through any generic point of the state space. For that purpose, all branches of roots that occur in the differential equation are considered, which is the same as finding the minimal-degree polynomial that contains the given equation as one case. The degree of the DE is then the degree of the highest order derivative. $y'=y^3$ and $y'=x^9$ are both degree $1$.

For a practical demonstration, when you put such an algebraic differential equation (not DAE) in WolframAlpha, it gives the solution in that extended context, which may be confusing if the given differential equation is real and a real solution is expected.

In real calculus terms, it is as you observed, you only add spurious solutions by switching to this smallest polynomial DE. The degree does not really serve a generally useful purpose here.

As for the example $y'=y^{1/3}$, under the definition of "polynomial in the highest order derivative, everything else is coefficient functions", the degree is clearly $1$. Under the definition "all roots dissolved, minimal polynomial in all derivative orders", you get $y'^3=y$ as this polynomial and thus degree $3$. The degree $9$ expression is not minimal as polynomial.