Question

2. Use the method for solving homogeneous equations to solve the following differential equation 8(x2 + y2)dx + 9xydy = 0 3. Solve the initial value problem y" – 4y' + 4y = 0, 17 y(0) = -3, y'(0) = 4

Answer 1

leur2 use homogeneous method to Rolve the following differential Equation, 8177+y?) dat My dy- 0 pal 817?ty) dat guy.dly gry dy=o. 811?+y?).d11. こ dy dr - 86x?tys) gay. put y= 12. V ndr dri da ey (11 an vt ndr vt adv Now pulling value of dy and value of yn -8177) dn 9 x (v) -8.72 (1+ve da 9x² v VoxdV 8 (1+r)] da xdv. - 8 [3+v) dh adu -8 f1+v?) 9V2 dn gv - 8 - 17 v? gv dv 9 v gv xe du 9V dan 7 8 +1712 Inegrating buth sides,

gv dv - 8717 v2 da 71 pust 8 117V dt = dv t 34V dt 34 rdv. 6 dt da sit 34 -lnxt 9 all terms lade constant c with In thon ha عليل + inc g dn (841702) = -lnx 34 Inf 2777v) + ARX ARC 34 put va عهای ln 8t17yz = nurt Inc. 34 22 In 934 8 +17 4/12 tuna= In c. Now, unat Inb= ln Cabl 10734 84 17 y 1912 In c. In. K f In a= lub than a b & the complete the complete Polution is. 9434 8+17 42 C X 72

Solvellu. iniliakalven parellem. yu- 1741.17.0, 460223, 40) = -17 46 giului put there values in the equal 10- 40+4)y 2 - 4D + 4 =0 12 - 2D - 20+4-0 DCD-22-2 (1-2) DE 2,2 - the complementary funcltun_10.9.) When Meals come quel is 27 y = (a +(ane? -> 91. Alush Lloy inepal fushamis. Complek solution is 2x ya (Citone of (186 by using Initial We will find life value Conditions at no, y=-3 -3= (Citole y(o)= -3.

17 4 Now opecond Conds kon is. golo)= 4164)= (( )Cºn+ 2 ({if(x) e?? at n=0 4--1714 G+ 21-3+ole -17 4 -17 - 12 ( 2-6 (2= -176 (G= 7 putting ralue of G82 Complete Polution. in equation of y get the g(x)= -3 + 7 4 7) pº y(x)= 7 X - 12 4 12