The provided differential equation is: \[ \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0. \]
Step 1: Algebraic manipulation of the equation. To ascertain the degree, eliminate the radical by squaring both sides. Rearrangement yields: \[ \sqrt{\frac{dy}{dx}} = 4\frac{dy}{dx} + 7x. \] Upon squaring both sides: \[ \left( \sqrt{\frac{dy}{dx}} \right)^2 = \left( 4\frac{dy}{dx} + 7x \right)^2. \] This simplifies to: \[ \frac{dy}{dx} = 16\left(\frac{dy}{dx}\right)^2 + 49x^2 + 56\frac{dy}{dx}. \]
Step 2: Determination of order and degree.
The order of a differential equation corresponds to the highest-order derivative present. In this equation, the highest derivative is \( \frac{dy}{dx} \), thus the order is \( \mathbf{1} \).
The degree is defined as the highest power of the highest-order derivative after the elimination of any radicals or fractional exponents. After the squaring operation, the maximum power of \( \frac{dy}{dx} \) is \( \mathbf{2} \).
Final Answer: \[ \boxed{1 \, \text{and} \, 2} \]