The Order of a differential Equation is the order of the highest derivative occurring in the equation.

The DEGREE of a differential equation is the degree (exponent) of the derivative of the highest order in the equation after the equation is freed from negative and fractional powers of the derivatives. 

For example, consider the differential equations written above. 

Example: Determine the order and degree of \(\cfrac{{{d^2}y}}{{d{x^2}}} + {\left[ {1 + {{\left( {\cfrac{{dy}}{{dx}}} \right)}^2}} \right]^{3/2}} = 0\) 

Solution: Rewrite the equation as follows: 

\({\left[ {1 + {{\left( {\cfrac{{dy}}{{dx}}} \right)}^2}} \right]^3} = {\left( {\cfrac{{{d^2}y}}{{d{x^2}}}} \right)^2}\)The equation has second order and second degree.

Example: \(y = 1 + x\left( {\cfrac{{dy}}{{dx}}} \right) + \cfrac{{{x^2}}}{{2!}}{\left( {\cfrac{{dy}}{{dx}}} \right)^2} + \cfrac{{{x^3}}}{3}{\left( {\cfrac{{dy}}{{dx}}} \right)^3} + ........\cfrac{{{x^n}}}{{n!}}{\left( {\cfrac{{dy}}{{dx}}} \right)^n} + ......\infty \)

Solution: Rewrite the equations as follows: 

\(y = {e^{x\cfrac{{dy}}{{dx}}}} \Rightarrow x\cfrac{{dy}}{{dx}} = \ln y\)

The equation is first order and first degree.

Example 2: Find the order and degree of the differential equation \(\sqrt {\cfrac{{{d^2}y}}{{d{x^2}}}}  = \sqrt[3]{{\cfrac{{dy}}{{dx}} + 1}}\)