Question

5. Consider the system of differential equations regarding x = c(t), y = y(t), and z = z(t): x' = 211 x + 212 y + 213 2 y = 221 x + 222 y + 223 2 z' = 231 2 + 232 y + 233 z where 211, 212,..., 233 are all real constants. Which of the following options could be a general solution of this system? (a) C C[-] é 2t + C2 eft (b) C-1 137 e-2t + C2 5 et (c) C +C₂ 5 e6t et + C3 (d) C + C2 5 et + COLA

Answer 1

system of differential 5. The given equations x = au + a22y + 923 Z y = 92, a +922 y + 923 Z z = 932 x + azzy +9332 where and, 012 933 are all peal constants. Here the coefficient matrix of the system ali a12 A= a23 A21 922 023 921 932 933 so we knon matrix Since А. 3x3 matrix A has 3 eigenvalues. that if Aį 3x3 then the general the system of the form xs det X2 ya Azt Z. solution of с, е dat be + 73 + ,е Y E2 Zm are where deg da ada eigenvalues of A, Xe day ya ere ( thosee 42 z the Linearly independent aigen vectors of A connes ponding to eigen values den dag do respectively, and ا و ريم are neal numbers.

option and general solution option (d) 11:3]: 1 are general solution of the given The general Solution given in option and (6) tion op is not of the form and nence the solution given in can not of the system 1 the solution given in 1 3 vectors not Linearly independent because 3 1/3 5 5 + 8 5 is expressed as a of the vectors vectors are . The solution given be 1 E ve the vectos 1 yo line an combination [] ㅗ 5 5 E Hence the line only dependent. in option (d) can not system.

a vecton v over is The set of vectors Y vectors deydz,--, dn} in space a field line aroly dependent dependent if and only it least one of the vectors of be expressed a linean combination semaining others. at the set can of the

For option Let das 1). [1] -- Let relation consider the where I, C, Cm 2 ada t C do +6 23 are Dear numbers and Da vectors in IR 3 the null Cada & 622 + 3 Qz Non - 4]--[?] --83-6 V > -7 +36 +9 -Cat56 G +56 +G This gives + 834 + 3 =0 - G+ 56 - +54 + 3 =0. This is homogeneous in Cl, Co. The of the system 3 system of equations coefficient determinant 1 1 5 5 1 1.5-34-1}+*-5+5} 8 (+0) has unique solution :. The system G =lz =0 Non cadet Lq + C d z = C = G = G =0

are This implies that da, Lag Lg linearly independent ve 3 E]: 5 5 pot are lineany independent. option is so :. The solution given im of the form the solution given in option general solution of the be could the given system. a