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.