Existence and Uniqueness. Under the assumptions of Theorem 1, we will prove that equ...

Existence and Uniqueness. Under the assumptions of Theorem 1, we will prove that equation (8) gives a solution to equation (4) on . (a,b) We can then choose the constant C in equation (8) so that the initial value problem (15) is solved.

(a) Show that since P (x) is continuous on (a,b) , Then µ (x) defined in (7) is a positive, continuous function satisfying

(b) Since

verify that y given in equation (8) satisfies equation (4) by differentiating both sides of equation (8).

(c) Show that when we let and choose C to be y0 µ (x0), the initial condition y(x0) = y₀ is satisfied.

(d) Start with the assumption that y(x) is a solution to the initial value problem (15) and argue that the discussion leading to equation (8) implies that y(x) must obey equation (8). Then argue that the initial condition in (15) determines the constant C uniquely.