The order and degree of the differential equation $[1+(\frac{dy}{dx}{)}^3{]}^{\frac{7}{3}}=7\frac{d^{2}y}{d{x}^2}$ are respectively:

To find the order and degree, the differential equation must be free from radicals and fractions in the power of the derivatives.Given: $[1+(\frac{dy}{dx}{)}^3{]}^{\frac{7}{3}}=7\frac{d^{2}y}{d{x}^2}$ Cubing both sides to remove the fractional power:
$$([1+(\frac{dy}{dx}{)}^3{]}^{\frac{7}{3}}{)}^3=(7\frac{d^{2}y}{d{x}^2}{)}^3$$
$$[1+(\frac{dy}{dx}{)}^3{]}^7=343(\frac{d^{2}y}{d{x}^2}{)}^3$$
Order: The highest order derivative present is $\frac{d^{2}y}{d{x}^2}$, so the order is $2$.Degree: The power of the highest order derivative is $3$, so the degree is $3$.Therefore, the order and degree are $2$ and $3$ respectively.