Degree of a differential equation

Degree is defined only when the equation is a polynomial in the derivatives y′, y″, y‴, …. It is the highest power (a positive integer) of the highest-order derivative.

Worked example: (y″)³ + (y′)⁵ + y = 0 has order 2 and degree 3. Only the power of y″ matters, not the 5 on y′.

Worked example: y″ + sin y = 0 has order 2 and degree 1. The sine is of y, not of a derivative, so the equation is still a polynomial in the derivatives.

Worked example: y′ + e^(y′) = x is not a polynomial in y′, so its degree is not defined. Its order is 1.

Clear radicals and fractions in the derivatives first. √(1 + (y′)²) = y″ squares to 1 + (y′)² = (y″)², so the order is 2 and the degree is 2.

Worked example: y = x y′ + 1/y′. Multiplying by y′ gives y y′ = x(y′)² + 1, order 1 and degree 2.

When they are defined, order and degree are both positive integers.