THEOREM 1 Existence and Uniqueness of Solutions Suppose that both the function f(x, y) and its partial derivative Dy f(x, y) are continuous on some rectangle R in the xy-plane that contains the point (a, b) in its interior. Then, for some open interval I containing the point a, the initial value problem 11. dy dx has one and only one solution that is defined on the interval I. In Problems 11 through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee uniqueness of that so- lution. dy dx = f(x, y), y(a) = b = 2x²y²; y(1) = − 1 (9)
THEOREM 1 Existence and Uniqueness of Solutions Suppose that both the function f(x, y) and its partial derivative Dy f(x, y) are continuous on some rectangle R in the xy-plane that contains the point (a, b) in its interior. Then, for some open interval I containing the point a, the initial value problem 11. dy dx has one and only one solution that is defined on the interval I. In Problems 11 through 20, determine whether Theorem I does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee uniqueness of that so- lution. dy dx = f(x, y), y(a) = b = 2x²y²; y(1) = − 1 (9)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:THEOREM 1 Existence and Uniqueness of Solutions
Suppose that both the function f(x, y) and its partial derivative Dyf(x, y) are
continuous on some rectangle R in the xy-plane that contains the point (a, b)
in its interior. Then, for some open interval I containing the point a, the initial
value problem
11.
dy
dx
has one and only one solution that is defined on the interval I.
In Problems 11 through 20, determine whether Theorem 1 does
or does not guarantee existence of a solution of the given initial
value problem. If existence is guaranteed, determine whether
Theorem 1 does or does not guarantee uniqueness of that so-
lution.
dy
dx
f(x, y), y(a) = b
=
: 2x²y²; y(1) = −1
(9)
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