JEE Main & Advanced Mathematics Differential Equations Definition

An equation involving independent variable \[x,\] dependent variable \[y\] and the differential coefficients  \[\frac{dy}{dx},\,\frac{{{d}^{2}}y}{d{{x}^{2}}},........\] is called differential equation.

Examples :

(i) \[\frac{dy}{dx}=1+x+y\]    

(ii) \[\frac{dy}{dx}+xy=\cot x\]

(iii)\[{{\left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)}^{3}}-4\frac{dy}{dx}+4y=5\cos 3x\]

(iv) \[{{x}^{2}}\frac{{{d}^{2}}y}{d{{x}^{2}}}+\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}=0\]

(1) Order of a differential equation : The order of a differential equation is the order of the highest derivative occurring in the differential equation. For example, the order of above differential equations are 1, 1, 4 and 2 respectively.

The order of a differential equation is a positive integer. To determine the order of a differential equation, it is not needed to make the equation free from radicals.

(2) Degree of a differential equation : The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions. The degree of above differential equations are 1, 1, 3 and 2 respectively.