Question

1 Pedro) Solve the following system of differential equations with initial value condition "+ 2x + y = 0 1 ++2y = 0 with (0) (0) - and (0) - (0) = -1

Answer 1

Given that differential equations x"+24+ 4y =o yut at 2y=0 here and y are dependent variables Let t is independent variable ginen equations can be written as o + 2*+ 4y =0 -- 0 So dy ta+2y= 0 .121 dt? we Solve it by method of elemination So en Us (2) become D'+2)* +4y = 0 - (3) (D?+2)y + x = 0 --() Now multiply equ (3) by (0²+2) and (4lby & 4 and Subtracting

(ཀྱ+ 2) + 4 ( ） +༢) ༔ 。 ༢(5+2)9 + ༢༥་ ང༠ (རྒྱ+ 2 j༡ -4༡༠ ( 2 - +) ༡༤༠ - (

Now Antilliary Equation is (m²+ 2)² - 4 = 0 ma+am 44-4 = 0 m² +4m²=0 m² (m²+4)=0 m, ao,M2 = 0, y = +2i, ma =-2 i so the solution of eqn is) to the cif of equisis C.F = (citanje o* + Cycoszyt +Casinzett solution of equis is given by mit ) = CF + P. I since p.L=0 a(t) = ctet + Cocos2t + asinat 1-16) given that a(o)=0 10 = ct .. (7) Now alt) = c2- 24 sinat + 24 Cosat given that allo) = -1 Ind= catala ... 131 Again differentiating with respect to it we get

W = - 4 Cosat atasinat Nowe putting values of n and no inegal) -4G cosat - 4 Casinad talentett G(osette Sinet) + 4y = 0 4 Y = 2 cosat taasinat 420, +2 t y (t) = -1 - Cat + cosat + ca sinat given that yo)=0 circg=g.. ( 19 ) Now by een (7) and (8) city=0 Siis3=0 26g=0 = 1=0 = 16,=0 so y(t) = -22 t + casinet Nowe differetenting with respect to it ylt) = -12 + Caloset given that y'(o) = -1 -1 = - + ca [244-(₂=-2 .. (10)

Now by eam (8) and (0) 2 lata = -1 214 -2=-2 4 C4 =-3 then 12 = 1/2 So we get Te,=0,13=0] 122='12). 164=-314 Hence [act) = 1 + a2 sinad ly(t) = -1 t - 3 sinet