Question

solve the homogenbosLS Linear system linear system of of the I folloing DER = x + x2 + 3x3 - 2x2 + 2x3. dx = 3X3

Answer 1

A) Given data- Homogeneous linear system of the following differential equations are = x,txat 3x3 dx de = 2x2 + 2x 3 de = 3X x = AX. A úl a materix defined by Coefficients x'- ri i 3 o 22 L 0 0 3 37 To 2 2 0 0 3 A A =

lising eigen values Solving the Linear System of differential equations det (A-AI) = 0 Глоо À l I = 0 1 0 AndI = [1- 3 0 2-3 2 Loo 3-| - 0 det of (A-II) = (1-) ((2-1) (3-1) - (0x2)) - (ox (3-7) – (0x2)) +3 (oxo - 0 (2-x)) = (1-1) (2-2) (3-1) - 0 -0 +3(0) = (1-1) (2) (3-1) det (A-XI) = 0 It means (1-x) (2-x) (3-1) = 0 These are eigen values

case 1) - x=1 (A-XI) k=0 (A-1I) k = 0.0 - O (A-II) ka.ro 137 0 0 2 E To I 375t equations are Ok, + 2 kq + 3 k₂ = 0 Ok, t kz tak, = 0 - Ok, toka + 2k₂ = 0. 2 substitute ky value in eq 2 k Valux we don't know so toking I k, = 1) so

Case (ii) - ^2=2 (A-2I) kara I 37k equations are -k, tkat 3 k z=0 - 3 2kg=0 [kg = 0. ea becomes -k, tka=0 a case fin 73 - 3 -2 1 37 (A-BI) ka lo 1 2 1 kg 0 0 0 kg

- 2141+k, 3k 3 - 0 > - , + 24320 → , - 2 old Id d - B 23, x C + ke +cke +

Ansiven