Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Show that x(t) is a solution of the IVP x' = f(x), x(0) = xo if and only if it satisfies the integral equation %3D x(t) = x0 + f(x(s)) ds.

Show that x(t) is a solution of the IVP x' = f(x), x(0) = xo if and only if it satisfies the integral equation %3D x(t) = x0 + f(x(s)) ds.

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.2: Remainder And Factor Theorems
Problem 51PS
See similar textbooks
icon
Related questions
Topic Video
Question

Kindly help me

Show that x(t) is a solution of the IVP x' = f(x), x(0) = x0 if and only if it satisfies the integral equation = Xo + f(x(s)) ds. Consider the sequence of functions as defined in the proof of the Existence and Uniqueness Theorem, i.e., Xo(t) = xo Xn+1(t) = xo + I f(xn(s)) ds, n = 0, 1, .... Show that if Xn(t) is defined, continuous and belongs to the compact set D for all t e [-a, a] and f e C'(D), then xn+1(t) is defined and continuous for all t E -a, a].
Transcribed Image Text:Show that x(t) is a solution of the IVP x' = f(x), x(0) = x0 if and only if it satisfies the integral equation = Xo + f(x(s)) ds. Consider the sequence of functions as defined in the proof of the Existence and Uniqueness Theorem, i.e., Xo(t) = xo Xn+1(t) = xo + I f(xn(s)) ds, n = 0, 1, .... Show that if Xn(t) is defined, continuous and belongs to the compact set D for all t e [-a, a] and f e C'(D), then xn+1(t) is defined and continuous for all t E -a, a].
Expert Solution
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Research Design Formulation
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra for College Students
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS