6. If af/dx is continuous on the rectangleR = {(t, æ) : 0 < |t – tol < a, 0 < |æ – xo] < 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.7. Define the sequence {un} byuo (t)= 20,Un+1 = x0 +to+f(s, un(8)) ds,п %3 1,2,....Use the result of the previous exercise to show that|f(t, un (t)) – f(t, un-1 (t))| < K|u,(t) – Un-1(t)|.

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Answered: 6. If af/ðæ is continuous on the…

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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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.