The separation is done in this form, as the differentiated variable u' has been separated from the other kind of variable(s) on the R.H.S.

This is similar to the definition of a separable diff eq where we write:

h(y) y' = g(x)

here too, the variable that involves the differentiation has been separated from the other variable.

Theorem 2 (Uniqueness). Suppose that both F(x, y) and \(∂F(x, y)/∂y\) are continuous functions defined on a region R as in Theorem 1.

Then there exists a number δ2 (possibly smaller than δ1) so that the solution y = f(x) to (), whose existence was guaranteed by Theorem 1, is the unique solution to () for x0 − δ2 < x < x0 + δ2.

The general first-order ODE is y0 = F(x, y), y(x0) = y0. __ (1)

We are interested in the following questions: (i) Under what conditions can we be sure that a solution to (1) exists? (ii) Under what conditions can we be sure that there is a unique solution to (1)? Here are the answers.

Theorem 1 (Existence). Suppose that F(x, y) is a continuous function defined in some region R = {(x, y) : x0 − δ < x < x0 + δ, y0 − ? < y < y0 + ?} containing the point (x0, y0).

Then there exists a number δ1 (possibly smaller than δ) so that a solution y = f(x) to (1) is defined for x0 − δ1 < x < x0 + δ1.

Notice how \(y_{1}\) is only a non-trivial solution of the complementary equation => \( y_{1}' + p(x)y_{1}' = 0\)––––(A)

We then speculate whether \(y = uy_{1}\) (with u as a constant) is a solution of the non-homogenous eqn. 2.1.16 or not.

To test this speculation, we put \(y = uy_{1}\) into 2.1.16 Which gives => \(uy_{1}' + u.p(x).y_{1}' = f(x)\)

But the L.H.S here is merely a multiple of (A) => L.H.S = 0 ≠ f(x)

Hence, the speculation that \(y = uy_{1}\) with u as a constant is a solution of 2.1.16 is rendered incorrect. u must instead be a function of x.