. Differential Equation, Order, and Degree Definition . A differential equation is any equation which contains at least one derivative of an unknown function, either ordinary derivative or partial derivative . For instance, let y ( ) where y is a dependent variable ( f is an unknown function) and x is an independent variable.
( ) The equation dy dx = is a differential equation. ( ) The equation dy = sin is a differential equation. Johann Bernoulli ( - ) - - ( ) The equation dy is a differential equation. ( ) The equation d y + = sin is a differential equation.
( ) The equation e x x dx = > log , is a differential equation. ( ) The equation tan − = d y dx is a differential equation. Definition . (Order of a differential equation) The order of a differential equation is the highest order derivative present in the differential equation.
Thus, if the highest order derivative of the unknown function y in the equation is k th derivative, then the order of the differential equation is k . Clearly k must be a positive integer. For example, d y d y − is a differential equation of order three. Definition .
(Degree of a differential equation) If a differential equation is expressible in a polynomial form, then the integral power of the highest order derivative appears is called the degree of the differential equation In other words, the degree of a differential equation is the power of the highest order derivative involved in the differential equation when the differential equation (after expressing in polynomial form) satisfies the following conditions : (i) All of the derivatives in the equation are free from fractional powers, if any. (ii) Highest order derivative should not be an argument of a transcendental function, trigonometric or exponential, etc. The coefficient of any term containing the highest order derivative should just be a function of x, y , or some lower order derivative but not as transcendental, trigonometric, exponential, logarithmic function of derivatives. If