What are the order and degree, respectively, of the differential equation \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2} = {{\rm{y}}^4} + {\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}?\)

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.

For example, if the equation is

\({\left( {\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}}} \right)^3} + \frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {{\rm{x}}^2} = 0\)

Here, order of the highest derivative is 2. And the degree is 3.

Calculation:

Given: \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2} = {{\rm{y}}^4} + {\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}\)

To find: Order and degree

So, the highest order is \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\)

∴ Order is 3.

Degree of differential equation is the degree highest order. So, degree is 2.