Question

10. Solve the system of differential equations by using eigenvalues and eigenvectors. x1 = 3x, + 2x2 + 2xz x2 = x + 4x2 + x3 X;' =-2x, - 4x2 – x3

Answer 1

foy? With the system of differentia equations in mateix form: X'(t): AX 3 2 2 X'(t) = 4 X - -4 Stop I and the eigenvalues of mattix A TA-II = 0 13-1 2 2 1 -0 4-X -4 1 -1- (3-2)[14-0)1-1-22+4)-2 [-1-2 +2] +2 [-4 +214-2)) = 0 (3-2)(x2-3x] -2(1-1) + 2 (-20+4) = 0 1 (3-4) (1-3) - 2 +21-4d + 8 =0 ^ (3-2)(8-3) + 6 - 2d=0 (3-1) [old-3) +21=0 (3-4) (1?- 32+2)=0 (3-d2 (1-1 (1-2)=0 The eigenvalues ose det, d2=2, 13 = 3 seep find the eigenvectons for each egionalne for 1,21 13-1 2 2 2. 2 1 4-1 2 1 1 1 -1-1 0 3 -2 -4 do -2 -2

solve the mately equation we get 1 O 0 21 22 - 0 0 0 0 0 0 0 xat azoa 20 4 = -13 -3 -1 " 12 0 3 ava 0 13 for d₂ = 2 3-2 2 2 2 2 2 1 4-2 - 0 0 1 2 -2 -4 -2 -4 -1-2 -3 0 O 0 solve the mabix equatión, 4++ 2 = -232 get 202=0 xzo 71 -2 -2 22 - 12 - 163 V2- Lo 0 0 for d₂ = 3 3-3 2 4-3 2 0 2 2 0 0 1 1 - - N 1 -2 -4 -1-3 0 1 0 0 -2 -4 -4 0 solve the mateix equation, we ger. 240 12 +13=0 * = -3

21 0 0 22 0 -1 13 -X3 * V3 = -1 3 1 lithe يقد Thus, the genual solution of system given by X(t)egedit vt hekt Vat ( erat 1-17 XCE)= get 0 + zesti t ce est V3 -1 1 0 -