The degree and the order of the differential equation \(\rm \dfrac{d^2y}{dx^2} = \sqrt{1+\left(\dfrac{dy}{dx}\right)^3}\) are:

Concept:

Order of a differential equation is the highest order of derivative that occurs in the differential equation.

Degree of a differential equation is the highest power of the highest order derivative that occurs in the equation, after all the derivatives are converted into rational and radical free form.

Calculation:

Getting rid of the radicals by raising both the sides to power 3 will give us:

\(\rm \dfrac{d^2y}{dx^2} = \sqrt{1+\left(\dfrac{dy}{dx}\right)^3}\)

\(\rm \Rightarrow \left(\dfrac{d^2y}{dx^2}\right)^2 =1+\left(\dfrac{dy}{dx}\right)^3\).

The highest derivative in it is \(\rm \dfrac{d^2y}{dx^2}\), therefore its order is 2.

And, the highest power of this derivative in the equation is also 2, therefore its degree is also 2.