Solution of Differential Equations
A solution of a differential equation is a relation between the variables , not involving the differential coefficients , such that this relation and the derivative obtained from it satisfy the given differential equation .
e . g ., Let d 2 y / dx 2 + y = 0
Integrating above equation twicely , we get y = A cos x + B sin x .
General Solution
If the solution of the differential equation contains as many independent arbitrary constants as the order of the differential equation , then it is called the general solution or the complete integral of the differential equation .
e . g ., The general solution of d 2 y / dx 2 + y = 0 is y = A cos x + B sin x because it contains two arbitrary constants A and B , which is equal to the order of the equation .
Particular Solution
Solution obtained by giving particular values to the arbitrary constants in the general solution is called a particular solution . e . g ., In the
previous example , if A = B = 1 , then y = cos x + sin x is a particular solution of the differential equation d 2 y / dx 2 + y = 0 .
Solution of a differential equation is also called its primitive .
Formation of Differential Equation Suppose , we have a given equation with n arbitrary constants f ( x , y , c1 , c2 ,…, cn ) = 0 .
Differentiate the equation successively n times to get n equations .
Eliminating the arbitrary constants from these n + 1 equations leads to the required differential equations .