Find Order And Degree of a Differential Equation

$$\left\{ 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right\} ^ { \frac { 3 } { 2 } } = \frac { d ^ { 2 } y } { d x ^ { 2 } } \tag 1$$ The highest order derivative is $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . So the order of the ODE is two.

In the definition of the degree, the key point is that the differential equation must be a polynomial equation in derivatives. The given differential equation is not a polynomial equation in its derivatives (because of the fractional power, 3/2, to which the term on the left hand side is raised) and so, strictly speaking, its degree is not defined.

However, in a less strict sense, the degree considered is the degree of the highest order derivative which is one. So the degree of Eq.$1$ is one, without forgetting that the definition of degree is not strictly respected.

If we take the liberty to transform the ODE so that all exponents be integers, $$\left( 1 + \left( \frac { d y } { d x } \right) ^ { 2 } \right) ^ 3 = \left(\frac { d ^ { 2 } y } { d x ^ { 2 } }\right)^2 \tag 2$$ the differential equation becomes a polynomial equation in its derivatives and the degree can be defined. Commonly, the degree considered is the degree of the highest order derivative. So, the degree of Eq.$2$ is two. (Without forgetting that Eq.$2$ is not strictly equivalent to Eq.$1$ ).