Different Translations(a) Show that for arbitrary a ≥ 0, y' = y has infinitely many so...

Different Translations

(a) Show that for arbitrary a ≥ 0, y' = y has infinitely many solutions that can be written

y(t)= e(t − a)

(b) Show that for arbitrary a ≥ 0, the IVP

has infinitely many solutions

(c) Sketch the similar graphs of the two families in (a) and (b). Explain in terms of uniqueness where and why they differ.

Picard Approximations In Problem 29 of Sec. 1.4 it was shown that the IVP

is equivalent to the integral equation

(Any solution of one is a solution of the other also.)This leads to the following.

Picard’s Successive Approximations

Beginning with a fairly arbitrary first approximation y0(t)(for example, y0(t)≡ y0. the constant function), we define a sequence of approximations y1, y2,…, yn, …using the formula

The proof of Picard’s Theorem depends on showing both that the sequence y0, y1, y2, … yn, … of functions tends to a limit function y(t), and that y is a solution of the integral equation, hence of the initial-value problem.