Let’s consider the 1st-Order linear differential equation
We see is a solution.
However, if (1) has a solution whose value at is not zero, it must be true that
where is a constant.
From (2), we obtain
where is a constant. It is either or .
We assert and prove that for any constant , a function defined by (3) is a solution of (1):
Notice when , (3) yields , the zero solution of (1).
Moreover, to see there are no other solutions, let be any solution of (1), we have
where is a constant. Hence, any solution of (1) belongs to the family of functions defined by (3)