Given below is the second-order system of differential equations: d2x/dt2 = -x - 3y, d2y/dt2 = -3x - y The coefficient matrix is as follows: [-1 -3] [-3 -1] It has eigenvectors [1] [-1] [1] and [1] and the corresponding eigenvalues are -4 and 2.
Given below is the second-order system of differential equations: d2x/dt2 = -x - 3y, d2y/dt2 = -3x - y The coefficient matrix is as follows: [-1 -3] [-3 -1] It has eigenvectors [1] [-1] [1] and [1] and the corresponding eigenvalues are -4 and 2.
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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Question
Given below is the second-order system of
d2x/dt2 = -x - 3y, d2y/dt2 = -3x - y
The coefficient matrix is as follows:
[-1 -3]
[-3 -1]
It has eigenvectors
[1] [-1]
[1] and [1]
and the corresponding eigenvalues are -4 and 2.
There are 5 options to choose from, please calculate the general solution
![[*] = C₁ [1] cos(24) + C₂ [¹] sin(2t) + C₂ [₁²¹] e¹² + 0₂ [₁¹] -√²
C3
e√²t C₁
-2t
[;)] = a[1¹] ~* + ¤ [1¹] - * - ¤ [i] *x{√/2t) + C+ [1] sin(√²2t)
e²t C₂
e + C3
C4
e²t C₂
-2t
+ a[¦] •*+a[1²¹] ²² - ¤ [₁¹] -√³
C3
√2t
+ C4
[i] = a[i]
H
[:] =^ [H]₁
cos(2t) + C₂
[i] sin (20) + C₂ [1¹] com({√/2t) + C₂ [1¹] sin(√/žt)
sin(2t) C3
[] = a[]* + a[i] + " + a[₁¹] cos(√28) + Cc [ ₁¹ ] sin (√/28)
H e
C₁
-2t
e²t
C3
C4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f7a3efc-b3ca-4051-8db3-5036d847891b%2F87c74427-c3ed-4801-9bd0-14f655c9115d%2Flnv954_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[*] = C₁ [1] cos(24) + C₂ [¹] sin(2t) + C₂ [₁²¹] e¹² + 0₂ [₁¹] -√²
C3
e√²t C₁
-2t
[;)] = a[1¹] ~* + ¤ [1¹] - * - ¤ [i] *x{√/2t) + C+ [1] sin(√²2t)
e²t C₂
e + C3
C4
e²t C₂
-2t
+ a[¦] •*+a[1²¹] ²² - ¤ [₁¹] -√³
C3
√2t
+ C4
[i] = a[i]
H
[:] =^ [H]₁
cos(2t) + C₂
[i] sin (20) + C₂ [1¹] com({√/2t) + C₂ [1¹] sin(√/žt)
sin(2t) C3
[] = a[]* + a[i] + " + a[₁¹] cos(√28) + Cc [ ₁¹ ] sin (√/28)
H e
C₁
-2t
e²t
C3
C4
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