In Chapter 1 we saw that an initial value problem of the form
dydx=f(x,y),y(a)=b
(1)
can fail (on a given interval containing the point x=a) to have a unique solution. For instance, in Example 4 of Section 1.3, we saw that the initial value problem
x2dydx+y2=0,y(0)=b
(2)
has no solutions at all unless b=0, in which case there are infinitely many solutions. According to Problem 31 of Section 1.3, the initial value problem
dydx=−1−y2−−−−−√,y(0)=1
(3)
has the two distinct solutions y1(x)≡1 and y2(x)=cos x on the interval 0≦x≦π. In this appendix we investigate conditions on the function f(x, y) that suffice to guarantee that the initial value problem in (1) has one and only one solution, and we then proceed to establish appropriate versions of the existence-uniqueness theorems that were stated without proof in Sections 1.3, 5.1, 5.2, and 7.1.