EBK ADVANCED ENGINEERING MATHEMATICS
EBK ADVANCED ENGINEERING MATHEMATICS
6th Edition
ISBN: 9781284127003
Author: ZILL
Publisher: JONES+BARTLETT LEARNING,LLC (CC)
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Chapter 6, Problem 5CR
To determine

The approximated value of y(0.2), for the equation y(2x+1)y=1, y(0)=3 and y(0)=1 with h=0.2 for one step and h=0.1 for two steps.

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Find the first four nonzero terms in a power series expansion about x=0 for a general solution to the given differential equation w''-14x^2w'+w=0
In this exercise, we will investigate a technique to prove that a language is notregular. This tool is called the pumping lemma.The pumping lemma says that if M = (S, I, f, s0, F ) is a DFA with p states (i.e., p = |S|) and if the wordw is in L(M ) (the language generated by M ) and w has length greater than or equal to p, then w may bedivided into three pieces, w = xyz, satisfying the following conditions:1. For each i ∈ N, xy^i z ∈ L(M ).2. |y| > 0 (i.e., y contains at least one character).3. |xy| ≤ p (i.e., the string xy has at most p characters).   Use the pumping lemma to show the following language is not regular (HINT: Use proof by contradictionto assume the language is regular and apply the pumping lemma to the language):L = {0^k1^k | k ∈ N}
A prefix of length ℓ of some word w are the first ℓ characters (in order) of w.1. Construct a context-free grammar for the language: L = {w ∈ {a, b}∗ | every prefix of w has at least as many a’s as b’s}2. Explain why every word generated by your context-free grammar (in Part 1) is contained in L. Then,prove via induction that every w ∈ L is produced by your context-free grammar.
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