Order and degree both describe a differential equation, but they measure different things. Order counts how many times the function has been differentiated in the highest term present; degree counts the power on that highest order term only, after clearing any roots or fractions involving derivatives.
Scanning the given equation, the term $\partial^3\varphi/\partial x^3$ carries the most derivatives, three of them, so the order is 3. Every other term, including the squared term $(\partial^2\varphi/\partial x^2)^2$, involves fewer derivatives (order 2), so that squaring does not enter the degree calculation at all, it only matters for a term sitting at the highest order.
Since the order 3 term shows up to the first power only, the degree is 1. That makes $m = 3$ and $n = 1$.
So $m - n = 2$, option (A).
Let \( y = f(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that \( f(0) = 0 \). If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then \( \alpha^2 \) is equal to ______.
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: