The order and degree of the differential equation $3-\left(\frac{\mathrm{d}^3 \mathrm{y}}{\mathrm{d} x^3}\right)^{\frac{7}{3}}=\left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)^5$ are respectively

To determine the order and degree of the differential equation $3-\left(\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}\right)^{\frac{7}{3}} = \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^{5}$, we identify the highest order derivative present.

The equation contains $\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}$ and $\frac{\mathrm{d} y}{\mathrm{d} x}$, making the order $3$.

To establish the degree, we isolate the radical term and eliminate fractional exponents by raising both sides to the power of $3$:
$\left[\left(\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}\right)^{\frac{7}{3}}\right]^{3} = \left[3 - \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^{5}\right]^{3}$
Thus, $\left(\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}\right)^{7} = \left[3 - \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^{5}\right]^{3}$.

The highest power of the highest order derivative is $7$, so the degree is $7$.