Special Equations are an extension of linear equations in two variables; where there is only one equation given and based on that one has to find the solutions satisfying the conditions. There can be exactly one solution or even more than that and the question might even ask the possible number of solutions for the same.
Let’s take an example:
Let us say we have an equation : x + 3y = 90
Now, this equation will have infinite number of real solutions. We can put a constraint to find number of non negative integral solutions for this equation.
The different solutions satisfying the equation are (0,30), (3,29),(6,28),……(90,0)
Now an interesting pattern follows here- the values of x satisfying the above equation are in A.P., with the common difference of 3, while the values of y are also in an Arithmetic Progression with a common difference of 1. The values of x are increasing from 0 to 90, while the values of y are decreasing from 30 to 0. Thus, there are in all 31 non negative solutions.
If we were asked the number of natural number solutions? The answer would be 29 as we will remove the two solutions-(0,30) and (90,0).