Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. uo (a) u' (t) = u(t) +3, u(0) = 3 (b) u' (t)=-u²(t) + sin(t), u(1) = 4 (c) u' (t) = 1/u(t), u(2) = 2 (d) u' (t)=ru(t) (1-u(t)/K), u(to) = uo (the logistic equation).

Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. uo (a) u' (t) = u(t) +3, u(0) = 3 (b) u' (t)=-u²(t) + sin(t), u(1) = 4 (c) u' (t) = 1/u(t), u(2) = 2 (d) u' (t)=ru(t) (1-u(t)/K), u(to) = uo (the logistic equation).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Do b and d

4 Exercises Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = uo has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. (a) u'(t) = u(t)+3, u(0) = 3 (b) u'(t)=-u²(t) + sin(t), u(1) = 4 (c) u'(t)=1/u(t), u(2) = 2 (d) u' (t) = ru(t) (1-u(t)/K), u(to) = uo (the logistic equation). oro very general form of Newton's law of cooling given by
Transcribed Image Text:4 Exercises Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = uo has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. (a) u'(t) = u(t)+3, u(0) = 3 (b) u'(t)=-u²(t) + sin(t), u(1) = 4 (c) u'(t)=1/u(t), u(2) = 2 (d) u' (t) = ru(t) (1-u(t)/K), u(to) = uo (the logistic equation). oro very general form of Newton's law of cooling given by
rectangle in the tu plane (in more advanced texts weaker conditions on f are permitted). We summarize these facts in the following theorem. Theorem 2.4.1 - Existence and Uniqueness of Solutions to ODES. Let R denote a rectangle a<t<b and c <u<d in the tu plane. Suppose that the function f(t, u) is continuous at each point in R and that is continuous at each point in R. Then for any point t = to, u = uo in R, the af ди ODE u' (t) = f(t,u(t)) has a unique solution with u(to) = uo on some interval to -8 <t<to + ₂ with 8₁> 0 and 8₂ > 0.
Transcribed Image Text:rectangle in the tu plane (in more advanced texts weaker conditions on f are permitted). We summarize these facts in the following theorem. Theorem 2.4.1 - Existence and Uniqueness of Solutions to ODES. Let R denote a rectangle a<t<b and c <u<d in the tu plane. Suppose that the function f(t, u) is continuous at each point in R and that is continuous at each point in R. Then for any point t = to, u = uo in R, the af ди ODE u' (t) = f(t,u(t)) has a unique solution with u(to) = uo on some interval to -8 <t<to + ₂ with 8₁> 0 and 8₂ > 0.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS