The order and degree of the differential equation $\sqrt{\frac{\mathrm{dy}}{\mathrm{d} x}}-4 \frac{\mathrm{dy}}{\mathrm{d} x}-7 x=0$ is respectively

The order of the differential equation is determined by the highest derivative present, which is the first derivative $\frac{dy}{dx}$ in this case.

To determine the degree, the equation must first be rationalized. We isolate the radical: $\sqrt{\frac{dy}{dx}} = 4\frac{dy}{dx} + 7x$, and square both sides, yielding $\frac{dy}{dx} = \left(4\frac{dy}{dx}\right)^2 + 2\left(4\frac{dy}{dx}\right)(7x) + (7x)^2 = 16\left(\frac{dy}{dx}\right)^2 + 56x\frac{dy}{dx} + 49x^2$.

The highest power of $\frac{dy}{dx}$ in the resulting expression is 2, so the degree is 2. The order is 1 and the degree is 2.

The correct option is A.