(d) Fact (i): It can be shown using part (c) that Fact (ii): It can further be shown using Fact (i) that | P(2)q(2) da = (p(x)d"(x) – p'(2)d(x) + p"(x)q(x) Let a be the last number of your student numbei, as usual. Using Fact (ii) and the table of values below, compute p(x)q(x) dr. You do not need to verify Fact (i) nor Fact (ii). p(a) p'(x) p"(r) q(x)(c) 4"(x) 1 5. 4. 1. 3. 9. 3
(d) Fact (i): It can be shown using part (c) that Fact (ii): It can further be shown using Fact (i) that | P(2)q(2) da = (p(x)d"(x) – p'(2)d(x) + p"(x)q(x) Let a be the last number of your student numbei, as usual. Using Fact (ii) and the table of values below, compute p(x)q(x) dr. You do not need to verify Fact (i) nor Fact (ii). p(a) p'(x) p"(r) q(x)(c) 4"(x) 1 5. 4. 1. 3. 9. 3
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
![And
3. Suppose p, q: [0, 1] R are three times differentiable on [0, 1] and that p"(r) = p(r) and
q" (1) = q(x) both hold for all a E [0, 1. Note: You do not need to provide reasons in part (c).
You'll discuss the reasons why we can use integration by parts in this problem in parts (a)-(b).
(a) Why are p" and " continuous on O, 1?
(b) Explain why .r
p"., and q .
are continnous on 0, 1.
(c) Using integration by par1s. verify that
Pgle) de
(7),()d
p'(x)q"(x) dr.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23c82962-fdd3-4aae-a199-ce5241f64aad%2F501f42bb-74c6-4dfa-9fd6-3b232634fc9b%2Fp1mktvp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:And
3. Suppose p, q: [0, 1] R are three times differentiable on [0, 1] and that p"(r) = p(r) and
q" (1) = q(x) both hold for all a E [0, 1. Note: You do not need to provide reasons in part (c).
You'll discuss the reasons why we can use integration by parts in this problem in parts (a)-(b).
(a) Why are p" and " continuous on O, 1?
(b) Explain why .r
p"., and q .
are continnous on 0, 1.
(c) Using integration by par1s. verify that
Pgle) de
(7),()d
p'(x)q"(x) dr.
Transcribed Image Text:(d) Fact (i): It can be shown using part (c) that
| P(@)q(2) de = (p(2)q"(=) - p ()d()+ "(=)d(2) de.
p(x)4"(x)-p'(x)d (x)
op (r) p(x),d
Fact (ii): It can further be shown using Fact (i) that
P(2>)a(n) dz = (pl2)d"(-) – -"(-)q(=)dr.
p(2)d(a)+p"(2)4()
"(2)q(x) dr.
Let a be the last number of your student numbei, as usual.
Using Fact (ii) and the table of
values below, compute
P(x)q(2) dr. You do not need to verify Fact (i) nor Fact (ii).
p(x) P(x) P() (2) 4() 4"(x)
(r) (r)
4.
1
8.
6.
Expert Solution
Step 1
Given, p,q:[0,1]→R are three differentiable functions. Also p'''(x)=p(x) and q'''(x)=q(x) for all x in[0,1]
(a) Since p'''(x)=p(x) , q'''(x)=q(x) and p(x) , q(x) are continuous so p'''(x), q'''(x) are continuous on [0,1]
(b) Since every differentiable function is continuous.
p(x),p'(x),p"(x), q(x),q'(x) and q"(x) all are differentiable so all are continuous.
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