The differential equation has infinitely many solutions. Solving a differential equation is referred to as integrating a differential equation since the process of finding the solution to a differential equation involves integration. A solution of a differential equation is an expression for the dependent variable in terms of the independent variable which satisfies the differential equation.
The solution which contains as many arbitrary constants is called the general solution. If we give particular values to the arbitrary constants in the general solution of the differential equation, the resulting solution is called a Particular Solution. The result of eliminating one arbitrary constant yields a first-order differential equation and that of eliminating two arbitrary constants leads to a second-order differential equation and so on. Let us understand solving the differential equation by an example.
(dy/dx) = x2y + y
Step 1: Divide the above differential equation by y. (We separate the variable)
(1/y)(dy/dx) = (x2 + 1)
We consider y and x both as variables and rewrite this as
(dy/y) = (x2 + 1)dx
Step 2: Now integrate L.H.S. with respect to y and with respect to x.
∫(1/y)dx = ∫(x2 + 1)dx
Step 3: After integrating, we get:
log y = (x3/3) + x + c
So, this is how the differential equation is solved.