Step 1: Define order and degree.
The order is the highest derivative that appears. The degree is the power of that highest derivative, but only after the equation is cleared of square roots and fractions in the derivatives.
Step 2: We want order 2, degree 3.
So the highest derivative must be $\frac{d^2y}{dx^2}$ (order 2), and once we clear the radical, its largest power should be governed by a cube structure giving degree 3.
Step 3: Examine option (D).
Option (D) is $\frac{dy}{dx} - \sin y = \left(\frac{d^2y}{dx^2}\right)^3\sqrt{\frac{d^2y}{dx^2}-1}$. The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$.
Step 4: Clear the square root.
Square both sides: $\left(\frac{dy}{dx}-\sin y\right)^2 = \left(\frac{d^2y}{dx^2}\right)^{6}\left(\frac{d^2y}{dx^2}-1\right)$. The radical is gone, which is the proper way to read the degree.
Step 5: Compare the other options.
Options (A) and (C) involve fractional or mismatched powers of the second derivative that do not settle into a clean degree 3, and (B) brings in a third derivative (order 3). Only (D) keeps order 2 while presenting the second derivative cubed as its defining power.
Step 6: Conclude.
Hence the equation having order 2 and degree 3 is option (D).
\[ \boxed{\dfrac{dy}{dx} - \sin y = \left(\dfrac{d^2y}{dx^2}\right)^3 \sqrt{\dfrac{d^2y}{dx^2} - 1}} \]