Question

find the degree of (d^2y/dx^2)^1/3=x^2+y dy/dx​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Find the degree of

$\sf \: {\bigg(\dfrac{ {d}^{2}y }{ {dx}^{2} } \bigg) }^{\dfrac{1}{3} } = {x}^{2} + y \: \dfrac{dy}{dx}$

$\large\underline{\bold{Solution-}}$

Basic Concept Used :-

Degree of differential equation :-

The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation. The differential equation must be a polynomial equation in derivatives for the degree to be defined.

CALCULATION :-

Given differential equation is

$\sf \: {\bigg(\dfrac{ {d}^{2}y }{ {dx}^{2} } \bigg) }^{\dfrac{1}{3} } = {x}^{2} + y \: \dfrac{dy}{dx}$

On cubing both sides, we get

$\sf \: \dfrac{ {d}^{2}y }{ {dx}^{2} } = {\bigg({x}^{2} + y \: \dfrac{dy}{dx} \bigg) }^{3}$

Since, higher order derivative exist in the given differential equation is 2 and its index is 1.

So,

Degree of differential equation is 1.

Note:-

The conditions to find the degree of differential equation is that the function should only be polynomial function, if the differential equation contains log, exponential and trigonometric function of the derivative then degree is not defined i.e. the equation has to be polynomial function to define the degree of a differential equation.

Additional Information :-

Order of differential Equation

The highest derivative in a differential equation is said to be a order of differential equation.