What is the degree of $y = x\frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$?

Explanation

1. 01

   Clear the root: $\left(y - x\frac{dy}{dx}\right)^2 = 1 + \left(\frac{dy}{dx}\right)^2$.

2. 02

   Expanding, $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} + y^2 - 1 = 0$: the only derivative is $\frac{dy}{dx}$, to the power 2, so the degree is 2.