Question

(b) X0 2 0 X with X(0)-3 0

Solve the initial value problem. Show all steps

Answer 1

be have to Solve χ1_AX okore A usith 0 2 o 3 0 Fiast vwe will hiod ut the eigen values of A and the coespand ergenvectons I4 I-2 1 I-A wh ere ¢ g the 3x3 identǐlgmataǐχ. t4 [o- (R-2]

→ Chara tous tic equa By solving this we get the eigenvalues of A. So eigen values of A are It remains to obtain the corresponding eievectos espunding ergenvetoas

Conn'de γ A-3, then A-31) a+b+4 a+ b-dc b 0 annedd b So eig en v eltovs a spanned b

02 Let aal,then vi = A+T) | b | = 0 21チ atb+4c 3 b =D

Henu the ergenvertors are spanned b ons an 2b Let b=1,then we get the eigen- vecton 2 V. 2. 0 Now, considet a=2 (A-21) b 0 -ㄏ oj

Summing tis toove get this tuoo, we equatin, n the above Substitu ting this 2 ors are So the set of eigenve Spanned b 2 2

Aue general oletion to Ax-x is aiven b 22t ast 2 Substinthe valius, e t 512 2 3 t +2t x3 at 2. 2

vex) Therefore, gue 2. 2. 2

Adding we get 44- lo 2 C22 So ουγ Solution S t -t at - at 2. 2.