The order and degree of the differential equation $\left(\dfrac{d^3 y}{dx^3}\right)^{\frac{2}{3}} - 3\dfrac{d^2 y}{dx^2} + 5\dfrac{dy}{dx} + 4 = 0$ are respectively

The given differential equation is $\left(\dfrac{d^3 y}{dx^3}\right)^{\dfrac{2}{3}} - 3\dfrac{d^2 y}{dx^2} + 5\dfrac{dy}{dx} + 4 = 0$.

The order of a differential equation is the order of the highest derivative present in the equation. The highest derivative is $\dfrac{d^3 y}{dx^3}$, so the order is $3$.

To find the degree, the differential equation must be expressed as a polynomial in its derivatives. Isolating the term with the fractional power, we get:

$\left(\dfrac{d^3 y}{dx^3}\right)^{\dfrac{2}{3}} = 3\dfrac{d^2 y}{dx^2} - 5\dfrac{dy}{dx} - 4$

Cubing both sides to remove the fractional power, we obtain:

$\left(\dfrac{d^3 y}{dx^3}\right)^2 = \left(3\dfrac{d^2 y}{dx^2} - 5\dfrac{dy}{dx} - 4\right)^3$

The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial in derivatives. The power of the highest order derivative $\dfrac{d^3 y}{dx^3}$ is $2$.

Thus, the order is $3$ and the degree is $2$.

Answer: $3, 2$