6. If af/ðæ is continuous on the rectangle R = {(t, x) : 0 < |t – tol < a, 0 < ]x – ¤o] < b} , prove that there exists a K > 0 such that |f(t, x1) – f(t, x2)| < K\x1 – x2| for all (t, æ1) and (t, x2) in R.
6. If af/ðæ is continuous on the rectangle R = {(t, x) : 0 < |t – tol < a, 0 < ]x – ¤o] < b} , prove that there exists a K > 0 such that |f(t, x1) – f(t, x2)| < K\x1 – x2| for all (t, æ1) and (t, x2) in R.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Exercise 6:
If ∂f/∂x is continuous on the rectangle
\[ R = \{ (t, x) : 0 \leq | t - t_0 | < a, 0 \leq | x - x_0 | < b \} \]
prove that there exists a \( K > 0 \) such that
\[ |f(t, x_1) - f(t, x_2)| \leq K |x_1 - x_2| \]
for all \( (t, x_1) \) and \( (t, x_2) \) in \( R \).
### Exercise 7:
Define the sequence \(\{u_n\}\) by
\[ u_0(t) = x_0, \]
\[ u_{n+1} = x_0 + \int_{t_0}^{t} f(s, u_n(s)) \, ds, \quad n = 1, 2, \ldots \]
Use the result of the previous exercise to show that
\[ |f(t, u_n(t)) - f(t, u_{n-1}(t))| \leq K | u_n(t) - u_{n-1}(t) |. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7e0ba88-0786-454c-b4b5-22240a9787f4%2Ff0d77027-92ab-437b-be5d-ff61eb095f73%2F7dr3bzc_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise 6:
If ∂f/∂x is continuous on the rectangle
\[ R = \{ (t, x) : 0 \leq | t - t_0 | < a, 0 \leq | x - x_0 | < b \} \]
prove that there exists a \( K > 0 \) such that
\[ |f(t, x_1) - f(t, x_2)| \leq K |x_1 - x_2| \]
for all \( (t, x_1) \) and \( (t, x_2) \) in \( R \).
### Exercise 7:
Define the sequence \(\{u_n\}\) by
\[ u_0(t) = x_0, \]
\[ u_{n+1} = x_0 + \int_{t_0}^{t} f(s, u_n(s)) \, ds, \quad n = 1, 2, \ldots \]
Use the result of the previous exercise to show that
\[ |f(t, u_n(t)) - f(t, u_{n-1}(t))| \leq K | u_n(t) - u_{n-1}(t) |. \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
