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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 80E
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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. 

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
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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