Question

Find the solution to the given system that satisfies the given initial condition.

Find the solution to the given system that satisfies the given initial condition. -7 -1 x(t) = x(t), 2-5 - 5 2 (a) (0) = (b) x(t) = (c) x( - 2) = 21 (d) (2) = 0 1 1

Answer 1

x'(t) = -7 -L X (+) be the given system of differential 2. -5 equation, or X'(+) = A x(+) Now characteristic equation of 1817-1 -7-x is 11* Gelto 1922 -5 -5.-2 (+) (5+X) +220 x2 +122 +35+2=0 -12 I 144-4.37 Oy X- 2 the IIA -61 , 2 let i = therefore two eigen values are -bri , 6-2 L:12=1 e] be the eigen vector correspondeng eigen value obti. AV = 66+i) V7 gives then where -7 -1 2 2 [3] 2 b on above matria of ③ then we apply elementary row operation I-i -i -1 Rg, t-i -(1+i)-1 A I- Retety Ry poo us therefore equavalent equation + tab=0 put b = then a=in. so Vya = 2 be an eigen vector for eigen values -6ti. the Solution corresponding to this eigen value and eigen veetor is

X(t) = (6+2)t e 79 est et [1 [ - i=-1 it [cost ti sen(t)] [121] .66 Cosct) sin(t) ti(cos(t) - since) 2 cos ct) i zsen(t) 792 - Cos (E)-Sinct) tiebt Cos(t)-sint ㄴ asint 2 cos (2) the general solution to this system then *&+)= e est -Cos (t) - Sin(t) 2 ees () COS(t) - Sin(t) 2 since)

Now put initial condition XCO) = 13 then [8]= [2]+ e[:] coso=t, Sinozo, e=1 - 5. X0= Is] is implies • 2¢0 7 F0 and e therefore the the solution with wrictial value X(t) cos(H - Since 2 sin(t) - best 1 X (H)= ther from * cost- sinh H-4én (sz.com min) cost -sinn + & eft 2 sint > cost=1, sin tico [1] = 42CM [] ***, [3] this gives eon G-e _-2.9 ET 60 - -2y egn een 2 and 6П - e e2= 60 a-eon ,677 - e en een een 2 So the solution is

X(t) 6t la cos(t) = sin(t) 169]-* =*[aunch Cos (t)-sin(t) 2 COSCL) X(-21) a pencatat [1] 1 - Setari then from Go. cos €217) - sin (-20) Aanr cos (2n) - sin (-25) tee ee selon [ [1] 2 Cos (-2.17) 2 slo(-21) o [] = get20 L21) & Get [:] CO(-) = CS this gives 4 2012 Bos (217) = a, sintag=0 = 1 es 12M, e and qeaza 120 121 t qe 2 =) -Ź e 127 e127) + qeder = 2 =h @etzn → 22 5² 12M, so the solution *c+) = čt21 esto fue es consiguen cos] + şeklame.com Cosct) . Sinct) 2 senit)

d X(72) = then from -6:57 & Ge [] (i Egers [cos27%) -im (90 ] Töltse cos) sin (2) 2 sin (2) como 2 Coscry) 317 qe -30 [2] Cosy= 0 sin (=d. this gives 29 e 39 = 1 -> @a e317 2 and -30 o -ą en + q e 30 -> → ea el therefore the => Eja solution is X(+)- 37 et cost-sen(t) e 2 zsin it)