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 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

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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