Definition:
An equation involving independent variables dependent variable and their derivatives is called a differential equation.
Ordinary differential equation:
A differential equation which involves only one independent variable is called an ordinary differential equation.
Order of a differential Equation:
- is the order of the derivative of the highest order, occurring in the differential equation.
Degree of a differential equation:
- is the degree of the highest order differential coefficient appearing in it, after all the differential coefficients are free from radical powers.
- To form a differential equation from a given equation in x, y and containing arbitrary constants. The given equation is differentiated successively as many times as the number of arbitrary constants. These equations are used to eliminate the arbitrary constants and the equation obtained is the required differential equation.
Solution of a differential equation:
- A functional relation between x and y which satisfies the given differential equation.
- Solution of a differential equation by the method of variables separable.
Step I :
Express the differential equation in the form f(x) dx = g(y) dy.
Step II :
Integrating both sides
Step I:
Put ax + by + c = t
Step II:
Substituting in the differential equation
Step III:
Reducing to variables separable
Step IV:
Integrating we get the solution.
Step III:
Substituting this in the differential equation, it reduces to the form f(v) dv = g(x) dx.
Step IV:
Solution is obtained by integrating both sides and substituting 
Step I:
Identify P and Q such that P and Q are functions of x only.
Step III:
Solution is obtained from
Step I:
Integrating once we get
is the solution of the given differential equations. 