Give the form of a particular solution of yl4) + 2 y " + 10 y "+ 18 y '+9y =5 e+ cos(x) +4 given that r1 = 3i is a root of the characteristic equation. a) O z-Ae*+ Bx cos(3 x) + Cx sin (3x) +D b) z=Axe*+Bcos(x) + C'sin (x) +D -X c) z=Ax?e*. +Bcos(x) + C sin(x) +D d) z=Ae*+Bcos(x) + C sin(x) + D e) z=Ax?e*. + Bx cos( 3 x) + Cx sin(3x) +D f O None of the above.
Give the form of a particular solution of yl4) + 2 y " + 10 y "+ 18 y '+9y =5 e+ cos(x) +4 given that r1 = 3i is a root of the characteristic equation. a) O z-Ae*+ Bx cos(3 x) + Cx sin (3x) +D b) z=Axe*+Bcos(x) + C'sin (x) +D -X c) z=Ax?e*. +Bcos(x) + C sin(x) +D d) z=Ae*+Bcos(x) + C sin(x) + D e) z=Ax?e*. + Bx cos( 3 x) + Cx sin(3x) +D f O None of the above.
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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![### Given Problem
**Determine the form of a particular solution for:**
\[ y^{(4)} + 2y''' + 10y'' + 18y' + 9y = 5e^{-x} + \cos(3x) + 4 \]
given that \( r_1 = 3i \) is a root of the characteristic equation.
### Multiple Choice Options
a) \( z = Ae^{-x} + Bx\cos(3x) + Cx\sin(3x) + D \)
b) \( z = Axe^{-x} + B\cos(x) + C\sin(x) + D \)
c) \( z = Ax^2e^{-x} + B\cos(x) + C\sin(x) + D \)
d) \( z = Ae^{-x} + B\cos(x) + C\sin(x) + D \)
e) \( z = Ax^2e^{-x} + Bx\cos(3x) + Cx\sin(3x) + D \)
f) \( \text{None of the above.} \)
### Explanation
For each option, the expression represents a possible particular solution for a non-homogeneous linear differential equation. The presence of exponential, trigonometric, and polynomial terms in these options is standard in solving such equations using the method of undetermined coefficients or variation of parameters.
The particular form of the solution is influenced by the roots of the characteristic equation and the terms on the right-hand side of the non-homogeneous equation. The root \( r_1 = 3i \) suggests complex conjugate roots, indicating oscillatory components in the solution. The exponential and trigonometric functions in the particular solution need to reflect the nature of these roots and the non-homogeneous terms \( 5e^{-x} \) and \( \cos(3x) \).
### Graphs or Diagrams
There are no graphs or diagrams in this content. Only equations and textual options are provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88afae02-80d5-49da-ad82-a7933e6f4445%2F75f18334-f917-4b42-8986-3d7796a7d6a7%2Fyykoisu_processed.png&w=3840&q=75)
Transcribed Image Text:### Given Problem
**Determine the form of a particular solution for:**
\[ y^{(4)} + 2y''' + 10y'' + 18y' + 9y = 5e^{-x} + \cos(3x) + 4 \]
given that \( r_1 = 3i \) is a root of the characteristic equation.
### Multiple Choice Options
a) \( z = Ae^{-x} + Bx\cos(3x) + Cx\sin(3x) + D \)
b) \( z = Axe^{-x} + B\cos(x) + C\sin(x) + D \)
c) \( z = Ax^2e^{-x} + B\cos(x) + C\sin(x) + D \)
d) \( z = Ae^{-x} + B\cos(x) + C\sin(x) + D \)
e) \( z = Ax^2e^{-x} + Bx\cos(3x) + Cx\sin(3x) + D \)
f) \( \text{None of the above.} \)
### Explanation
For each option, the expression represents a possible particular solution for a non-homogeneous linear differential equation. The presence of exponential, trigonometric, and polynomial terms in these options is standard in solving such equations using the method of undetermined coefficients or variation of parameters.
The particular form of the solution is influenced by the roots of the characteristic equation and the terms on the right-hand side of the non-homogeneous equation. The root \( r_1 = 3i \) suggests complex conjugate roots, indicating oscillatory components in the solution. The exponential and trigonometric functions in the particular solution need to reflect the nature of these roots and the non-homogeneous terms \( 5e^{-x} \) and \( \cos(3x) \).
### Graphs or Diagrams
There are no graphs or diagrams in this content. Only equations and textual options are provided.
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