Question

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I AM CURRENTLY IN DIFFERENTIAL EQUATIONS

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Consider the system of differential equations regarding r = r(t), y = y(t), and z = z(t): I' = 011 3 + 212 y + 213 z y' = 221 x + 222 y + 223 2 z = 231 x + 232 y + 233 2 where 211, 212, .. , 033 are all real constants. Which of the following options could be a general solution of this system? 1 + C2 ܟ ܕܢ e6t (a) C (-1)-2+ 1 (b) C 1 e-2t + C2 5 5 [1 + C2 5 e6t + C3 0 (c) C . 1 (d) C1 3 + C2 5 e6t + C3 5 1-+ 0

Answer 1

given system of differential equations x = aux + 9 12 y + 9 3 Z y = 0212 + 922 y + 923 Z z'- 931 x + 932y +9332 where all, 9121 933 arre all peal comtants. coefficient matrix of the system Here the aus 0 a 12 A= a 13 azt 022 023 021 032 Since A matrix 30 A has knon 3x3 3 eigenvalus. that + A is 3x3 the general solution of of the form matrix x2 the system det с. е yu 2. dat the Azt + се Y Zz are de, de 2 d3 eigenvalues of A, where xe xy x2 are thaoree Ч. z :( 2 23 22) linearly independent eigen vectors of A conses ponding to the eigen values de, dz, de respectively, and numbers. وه رد are scal

option a option and general solution of I vectors 1]:] [:] are general solution of the given The general solution given in option and is not of the form and nence the solution given in can not be the system For the solution given in option (d) 3 the not Linearly independent because 3 1 / 3 5 + is expressed as the Hence the of vectors line only dependent. . The solution given in option (d) can not :] ve the vector 5 1 line an combination to vectors [?] E] are System

oven a field F set of vectors ? 229d2,-., dn} in Necton space linearly dependent if and only if at least one of the vectors of the set be expressed a linean combination of pemaining others. For option ci 1 3 let A= 5 5 ise 1 A is the matrix formed by the vectores 1],[1742 will be linearly independent Non the vectors if and only if det A 70. det A = 1.5-34-17 +22-5+57 +0. The vectors A]:) 1 care solution lineanly als independent. The given in option of the form o So the solution in option could be a general solution of the given system, given