Solve the following homogeneous system of first order ODE: dx = 6x - 4y; dt dy = 4x - 2y dt with initial conditions: x(0

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Solve The Following Homogeneous System Of First Order Ode Dx 6x 4y Dt Dy 4x 2y Dt With Initial Conditions X 0 1 (74.29 KiB) Viewed 27 times

Solve the following homogeneous system of first order ODE: dx = 6x - 4y; dt dy = 4x - 2y dt with initial conditions: x(0) = 1, y(0) = 2. (10 marks) Hints: For the three cases of eigenvalues of A, where y' = Ay: Case 1 involves distinct eigenvalues (2) of A, and the general solution is: y = C1x(1) edit + czx(2) Azt + ... + Cnx(n)eint where x(n) are the corresponding eigenvectors. Case 2 involves Hermitian matrix A with repeated eigenvalues of multiplicity m, and the general solution is: y=c7x(1) eat + ... + Cmx(melat + Cm+1*(m+1) e Am+10 + ... + cnx(n)eint Case 3 involves non-Hermitian matrix A with repeated eigenvalues of multiplicity m, and the general solution is: y = cıya (1)emat + ... + Cmyam)elat + Cm+1+(m+1) eam+16 + ... + Carneant and = 0 = (A – 1,1)x(1) (A – 1,1)x2 (1) Ex : (A - 121)x(m) (m-1) = Xa ya = EX (1) plat Уа = Xa tenat + x2 erat : tm-1 yem) = x1 ya - eat txa = tm-2 elat + ... + xmetat () (m - 2)! (m - 1)!