HI2. Using D = as a differential operator, and using the following identity (D - a)(D – B) f(x)ed = e"(D+1- a)(D + A – B)f(x) A. Show that 1) (Aa² + Bx + -c)o*] = (iAz² ic)e= (D² + D +1) + [iB + (4i + 2)A]x + 2A+ (2i + 1)B + B. Use you result in A to find the particular solution for y" + y' + y = xečx C. From the real and imaginary parts of your result for B, find the particular solution to the following equations: (i) y" + y' + y = x cos(x) (ii) y" + y' + y = x sin(x)
HI2. Using D = as a differential operator, and using the following identity (D - a)(D – B) f(x)ed = e"(D+1- a)(D + A – B)f(x) A. Show that 1) (Aa² + Bx + -c)o*] = (iAz² ic)e= (D² + D +1) + [iB + (4i + 2)A]x + 2A+ (2i + 1)B + B. Use you result in A to find the particular solution for y" + y' + y = xečx C. From the real and imaginary parts of your result for B, find the particular solution to the following equations: (i) y" + y' + y = x cos(x) (ii) y" + y' + y = x sin(x)
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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![d
HI2. Using D
as a differential operator, and using the following identity
dx
(D – a)(D – B) f(x)e^ = e^# (D+ A – a)(D+A – B)f(x)
A. Show that
(D² + D+ 1)|(Aæ² + Bx +
+ Bx + C)e** = (iAa²
+ [iB+ (4i + 2)A]x +2A+ (2i + 1)B+ iC)e*
B. Use you result in A to find the particular solution for y" + y' + y
xeir
C. From the real and imaginary parts of your result for B, find the particular solution to the following
equations:
(i) y" + y' + y = x cos(x)
(ii) y" + y' + y = x sin(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40ea71b2-bf31-450c-ab0b-fcc43526c261%2F7b2311b1-5046-49fa-bd52-17d85b3cfd5d%2Ft9bjhvx_processed.png&w=3840&q=75)
Transcribed Image Text:d
HI2. Using D
as a differential operator, and using the following identity
dx
(D – a)(D – B) f(x)e^ = e^# (D+ A – a)(D+A – B)f(x)
A. Show that
(D² + D+ 1)|(Aæ² + Bx +
+ Bx + C)e** = (iAa²
+ [iB+ (4i + 2)A]x +2A+ (2i + 1)B+ iC)e*
B. Use you result in A to find the particular solution for y" + y' + y
xeir
C. From the real and imaginary parts of your result for B, find the particular solution to the following
equations:
(i) y" + y' + y = x cos(x)
(ii) y" + y' + y = x sin(x)
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