Step 1: Rewrite with a shorter notation.
Let $p=\frac{dy}{dx}$ and $q=\frac{d^{2}y}{dx^{2}}$. The equation is $\sqrt{1+p^{2}}=q^{1/3}$.
Step 2: Raise to the sixth power.
The powers $\frac{1}{2}$ on the left and $\frac{1}{3}$ on the right need a common multiple, which is 6. Raise both sides to the power 6.
\[ (1+p^{2})^{3}=q^{2} \]
Step 3: Read off order.
The highest derivative in this polynomial equation is $q=\frac{d^{2}y}{dx^{2}}$, so the order is 2.
Step 4: Read off degree.
The power of $q$ is 2. The other side, $(1+p^{2})^{3}$, only involves the first derivative and does not change the degree. So the degree is 2.
Step 5: Eliminate options.
Option 2 gives a fractional degree, which is never allowed once the equation is cleared. Option 3 says degree is not defined, but the equation is polynomial in derivatives, so degree exists. Option 4 uses degree 3, which is the power on the left bracket and not on the highest derivative.
Final Answer:
Option 1 is correct. \[ \boxed{\text{order}=2,\ \text{degree}=2} \]