Determine the degree and order of the given differential equation respectively. \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)^{-2}}?\)

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.

Note:

• Degree is defined if the function is a polynomial, if differential contains the logarithmic, exponential and trigonometric function of the highest derivative, then a degree is not defined.
• Degree and order are always positive integers.

Calculation:

Given: \({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + {\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)^{-2}}\)

To find: Order & Degree

\({\rm{y}} = {\rm{x}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + \frac{1}{{{{\left( {\frac{{{\rm{dx}}}}{{{\rm{dy}}}}} \right)} ^2}}}\)

\(y = (x +1)(\frac{dy}{dx})^2\)

Hence, the degree is 2 & order is 1.