Let a differential equation be expressed as a polynomial equation in the derivatives occurring in it.
The exponent of each of the derivatives should be minimum. Exponent of the variables need not be an integer.
Under such conditions,
i) the order of the differential equation is the order of the highest derivative in it.
ii) the degree of the differential equation is the largest exponent of the highest order derivative.
Note: If the differential equation cannot be expressed as a polynomial of derivatives, its 'degree' is not defined.
These are explained by the following examples.
It is polynomial in , and .
is the highest order derivative.
Hence the order of the differential equation is 3.
Now, the exponent of is 2, that of is 1 and that of is 3.
For finding the degree, we have to consider the exponent of highest order derivative which is of third order () in the given equation. Its exponent (or power) is 2.
Therefore, the order of the given differential equation is 3 and its degree is 2. Hence the degree is 2.
The general form of an ordinary differential equation of nth order is
or F(x, y, y1, y2, .... yn) = 0
or F(x, y, y', y'', ..... y(n)) = 0
Note that, it is not mandatory that derivatives of all orders need to be present in a differential equation.