Q14: Order and degree of $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = \frac{d^2y}{dx^2}$

The order of a differential equation is the order of the highest derivative in it. Here the derivatives are $\frac{dy}{dx}$ (order 1) and $\frac{d^2y}{dx^2}$ (order 2).

The degree is the highest power of the highest order derivative, once the equation is a polynomial in the derivatives. No derivative here sits inside a root, a log or a trig function, so the degree is defined.

Watch out: the power 3 belongs to the bracket with the first derivative. Expanded, its highest term is

but the first derivative is not the highest order derivative, so its power does not decide the degree. That rules out (B) and (D).