The order and degree of the differential equation $\frac{d^{3}y}{d{x}^3}=4\sqrt{{\left(\frac{dy}{dx}\right)}^3+y^2}$ are respectively:

The given differential equation is:
$$\frac{d^{3}y}{d{x}^3}=4\sqrt{{\left(\frac{dy}{dx}\right)}^3+y^2}$$
To find the degree, we must first express the equation in a form free from radicals as far as derivatives are concerned. Squaring both sides:
$${\left(\frac{d^{3}y}{d{x}^3}\right)}^2=16\left[{\left(\frac{dy}{dx}\right)}^3+y^2\right]$$
The order is the highest derivative present, which is $3$ (from $\frac{d^{3}y}{d{x}^3}$).The degree is the power of the highest derivative, which is $2$.Therefore, order = $3$ and degree = $2$.