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

i.e.,

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

.

Therefore,

.

That is

or,

where is a constant. Hence, any solution of (1) belongs to the family of functions defined by (3)

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