degree of a differential equation
Order tells you how deep the differentiation goes; degree is the partner label that tells you how nonlinearly the highest derivative enters. If order is the floor a building reaches, degree is roughly how the top floor is shaped. The two together are the classic opening description of an equation: 'first order, first degree', and so on.
The degree is defined only after the equation has been cleared of fractional and irrational powers of its derivatives — written as a polynomial in the derivatives. The degree is then the power to which the highest-order derivative is raised. So dy/dx = x + y is first degree; (dy/dx)^2 + y = 0 is second degree; and d^2y/dx^2 + (dy/dx)^3 = x is first degree (the highest derivative, the second one, appears to the first power — the cube is on a lower derivative and does not count). If the equation cannot be written as such a polynomial at all — for instance dy/dx = sin(dy/dx) or it contains the derivative inside a non-polynomial function — the degree is undefined.
In practice degree is a far less load-bearing label than order, and a working analyst rarely needs it. It matters mostly for a clean classification and for one honest fact: an equation of first degree in all its derivatives is exactly a linear equation only when the coefficients do not involve y. Degree one is necessary but not sufficient for linearity — which is why 'linear versus nonlinear' is treated as its own, more useful distinction.
The Clairaut equation y = x*(dy/dx) + (dy/dx)^2 is first order (only dy/dx) but second degree (that derivative is squared). The eikonal-type relation (dy/dx)^2 = 1 + y is also first order, second degree.
Order and degree are independent: an equation can be low order yet high degree.
Degree requires the equation to be polynomial in the derivatives first; if it is not — e.g. a derivative sits inside sin, log, or under a non-perfect root that cannot be cleared — the degree is simply undefined, and the question is not worth forcing.