Question

This is a differential equations course, please answer all the questions.

Find the general solution.

12. z'(t) -1)=(0) 13. z'(t) = -3)=(0) 14. zº(t) = (+5 -1)=(e) 5 10 0 15. x'(t) = 21 -2 r(t) 3 2 -1) 16. z'(t) = -30 2 1 -1 0 (t) -2 -1 -1 0

Answer 1

SOLUTION 72) ' (t) = 2. I Mih nino X(t) r(t) = AX(+) Eigen value equation for Coefficient matrix /A-AT/=0 Q- --- ule -/- (2-x) (-1-2) + 5 x 32 = 0 x2-x-2+1=0 12 -d-/to d = 1 Vite = 1355 Roots/ Eigen values are 2,= !+V5; d=1-15 2 Now Eigen vector corresponding 1,= 17V5; [A-1,13% 2- (+15) .5 ( x = 16 WIN [:] -1- (1485) 3-15 2 ale 1[:] [:) -3-15 Z

R2 R2 - RI 5(3-5) 3-85 2 - 2 7[*]+[:] Here 22 is free variable Choose X2 = 1 3-5 2 x 24 Ş X2 = 0 2 / 2 3-85 5 ( 3+15) (3-15) (3+15) 5 ( 3+15) 21 Hence eigen vector X - (6): [ 2017) . Now eigen vector corresponding 122 - 3 / 3 1-85 2 2 (4-05) -V5 5 -) - 12 13

3+ √5 - 24 [20 -3+VE 2 R2 R2- 5 (3+15) STAR so :] [2][ ~ 3+v5 2 Here X is free variable choose X2=1 3+05 2 21 E x2 = 0 Z 5 3+15 х 3-V5 3-15 こ 5 ( 3-15) Henee eigen vector X2 = (3-15)

Hence complele Solution: mit + cx xr ett X (t) = ex, e za se (1+5)* X (t) { (3+V5) 7 e (-V5 ) + **(****) 2170

Solved only 12)