Example 1.2.3 For some constant A, solve: y' = y', y(0) = A. Show solution Solution We know how to solve this equation. First assume that A # 0, so y is not equal to zero at least for some x near 0. So x'=1 7 2 , SO x = - + + C , so y = 7 1 -. If y(0) = A, then C = 1 - so A y = - I If A = 0, then y = 0 is a solution. For example, when A = 1 the solution "blows up" at x = 1. Hence, the solution does not exist for all x even if the equation is nice everywhere. The equation y'= 32 certainly looks nice. For most of this course we will be interested in equations where existence and uniqueness holds, and in fact holds "globally" unlike for the equation y'= 3. ? Exercise 1.2.4 Recall from Exercise 1.1.3 that y = VC-x 1 is a solution to y'= 2 . . If y(0) = A, for which a does the solution "blow up"? Hint Question Help: Message instructor D Post to forum Submit Question Next Page Ononlinear example we have seen previously, y'= y (Example 1.2.4), and to Picard's theorem (Section 1.2.4). ? Exercise 1.4.2 What do you get if you plug co in for @ in the definite integral form? y(20) = Question Help: Message instructor D Post to forum Submit Question Next Page O