The order and degree of the differential equation \(\rm k \dfrac{dy}{dx}=\displaystyle\int \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\) are respectively

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Here, \(\rm k \frac{dy}{dx}=\int \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx\)

Differentiation with respect to x, we get

\(\Rightarrow \rm k \frac{d^2y}{dx^2}=\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{2}{3}}\)

Taking cube on both sides, we get 

\(\Rightarrow \rm k^3 \left( \frac{d^2y}{dx^2}\right)^3=\left[1+\left(\frac{dy}{dx}\right)^2\right]^2\)

So order = 2 and degree = 3

Hence, option (2) is correct.