Solution of a Differential Equation
A solution of a differential equation is an equation which contains arbitrary constants as many as the order of the differential equation and is called the general solution. Other solutions, obtained by giving particular values to the arbitrary constants in the general solution, are called particular solutions.
Also, we know that the general integral of a function contains an arbitrary constant. Therefore, the solution of a differential equation, resulting as it does from the operations of integration, must contain arbitrary constants, equal in number to the number of times the integration is involved in obtaining the solution, and this latter is equal to the order of the differential equation.
Thus we see that the general solution of a differential equation of the nth order must contain n and only n independent arbitrary constants.