Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

APPENDIX A Existence and Uniqueness of Solutions

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

\frac{d y}{d x} = f (x, y), y (a) = b

$\frac{d y}{d x} = f (x, y), y (a) = b$ (1)

can fail (on a given interval containing the point $x = a$ $x = a$ ) to have a unique solution. For instance, in Example 4 of Section 1.3, we saw that the initial value problem

x^{2} \frac{d y}{d x} + y^{2} = 0, y (0) = b

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

has no solutions at all unless $b = 0,$ $b = 0,$ in which case there are infinitely many solutions. According to Problem 31 of Section 1.3, the initial value problem

\frac{d y}{d x} = - \sqrt{1 - y^{2}}, y (0) = 1

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

has the two distinct solutions $y_{1} (x) \equiv 1$ $y_{1} (x) \equiv 1$ and $y_{2} (x) = \cos x$ $y_{2} (x) = \cos x$ on the interval $0 ≦ x ≦ π .$ $0 ≦ x ≦ π .$ 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.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for APPENDIX A Existence and Uniqueness of Solutions

Create new playlist

Sign In

Sign Up

Table of Contents for
APPENDIX A Existence and Uniqueness of Solutions