Order of a differential equation

order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

Consider the following differential equations:

The equations (6), (7) and (8) involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively.

Degree of a differential equation

To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider the following differential equations:

We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10) is a polynomial equation in y′ (not a polynomial in y though). Degree of such differential equations can be defined. But equation (11) is not a polynomial equation in y′ and degree of such a differential equation can not be defined.

By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.

In view of the above definition, one may observe that differential equations (6), (7), (8) and (9) each are of degree one, equation (10) is of degree two while the degree of differential equation (11) is not defined.

NOTE: Order and degree (if defined) of a differential equation are always positive integers.

Example Find the order and degree, if defined, of each of the following differential equations:

Solution

(i) The highest order derivative present in the differential equation is , so its order is one. It is a polynomial equation in y′ and the highest power raised to is one, so its degree is one.

(ii) The highest order derivative present in the given differential equation is , so its order is two. It is a polynomial equation in and and the highest power raised to is one, so its degree is one.

(iii) The highest order derivative present in the differential equation is y′′′ , so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined.

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