Differential Equations and Linear Algebra, 4th Edition

APPENDIX A Existence and Uniqueness of Solutions

In Chapter 1 we saw that an initial value problem of the form

$$\frac{dy}{dx}=f(x,y),y(a)=b$$ (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

$$x^2\frac{dy}{dx}+y^2=0,y(0)=b$$ (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

$$\frac{dy}{dx}=-\sqrt{1-y^2},y(0)=1$$ (3)

has the two distinct solutions $y_1(x)\equiv 1$ and $y_2(x)=\cos x$ on the interval $0\leqq x\leqq \pi .$ 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 ...