Question

Problem point Consider the new system and the grand eigen vector for the court , and- Finder on the room 3y + 2y --- (0) (0) -- Use the win your - Note: canon this pro Problem por Find the most per real valued on to the line system of differential equations?

Answer 1

problem 5 å Consider the Linear system 3'=1327 3 -5-3 ] find the vectors eigen value and eigen for the coefficient matrix. Q=0, and C2:00 - -5-3 32 A= the given system %. $ 3) the coefficient matrix is and no != 3) 3 Gharacteristic equation IA-AI= 0 5-3 3 = ) the =) 3-a 2 -5-3- +10=0 - 5) - (3-1) (3+) 7-09-17) +10 - 0 4-9 +10 12+1=0 =) ů =+ therefore the eigen value aici, A2= -1 thus the eigen value aia i coeresponding eigen AV = ari rector v = (62) then

therefor (A-2) a = 0 -) 3-i 2 [s (3) -3-1 2 (3-1) + 2 U₂ = 0 -501 + (-3-1) V2 = 0 choose V=2 I =-3ti 2 -3ti therefore the eigen vector vi similarly -ing eigen vectors 12 = (v) then the eigen value aa = -1 and coeresponding AV2 - Q V2 2) (A -22) v2 = 0 3 ti 2 = ) -5 -3ti (3+i)v' +2 v' = 0 - svi + (-3 tij u = v=2 then '= -3-1 choose 2 therefore eigen vector V2 = (-3 Henee a = i, u 2 -3ti and a2 = -i, v = (-3-1) (132) (Ang). 6 find the real valued solution to the Initial value problem s mi= 39, +292 9, (o)= 6 = -54,-342 92 (0) = -5

=) -54,- 372 132 A= is and U2= where ie si = 24 cost + 2q sint the given system mi- 34,+29 the Coefficient me natrix coresponding the eigenvalue is a = i, 12 = – i and eigen vector Vi= (3+1) (-31) [privious solue (2) therefore the general solution as, + ₂ S2 eot $12 sint 3 Cosz- sinz sint t 2 sint - 3 sing + cost (asint -3 Cosz-sint - sinz tost teost) Henee milt 92(t) 9(-3603 t -sint) +cz (-38in+tcost) use the condition y(0) = 6 2.9 cos(o) +222 sin(8) 2) 9 = 1 / 2 - 4 = 3 Y(0) = -5 thus -5= 91-3-0) + (041) and S2 et? (3) (o jest iesz therefore y(t) = 2cost 4 (-34 th - 3 And

- ) - 5 = – 39 te C2 = – 5 +39 =) - - 5+ 3x3 (9-3) - 5+9 = -2 C2 = 4 therefore the required solution gilt = 2x3 Cost +2x4 sint 6 cost + 8 sint (Ang). 3(-3 cost - sint ) + 4 (-3 Sint + cost 2) Yit And - 5 cost-15sint 9. yolda ie Yolu = 1-5( cost +3sint) (AMS) problem 6 :- ă 'a (-2 - 4 I a the given differential equation the coefficient matrix h = (*_)? 2 - Comedy А. IT and the charaeteristic Equation 1 A - dil=0 -2-7 ~ - 4-7 Do 3) 1762 +9=0 (2+3) 0 -) = -3,-3,

eigen value a-3, Q2 = -3 which is repeated eigen the eigen value a=-3 corrsponding eigen vector e then Now AC =ae - (A-2) e=0 -2+3 l 1 y - -4+3 2 ) 1 a ei P ) eteg to - enero Choose ez=-| then e = 1 A rector ot therefore the eigen vector e = (-1) generalized eigen g corisponding eigen Ag (A-22) q (4) en value Az =-3. agt =) {& 1 1 T -1 2 ) 91 +92=1 - g -92 = -1 Now we choose 91-1 then therefore the eigen vector is 82=0

Hence the Required General solution est (D) : [< (4) +c7(4)t + (8)}]2 they ay(t) = -3t (6 + (t+1] eze and h(t) (-9 - Gt) est therefore (t) H2 Lt 1.83t (th)ě34] - a + - ezt -test (Ans).