Question

Find an integrating factor of the form X"y" and solve the equation. (2x-172-9y)dx + (3y-6x) dy=0, y(1) =1 OA 4x2y3 – 3x3y2 = 1 08.3x2y3 – x3y2=2 ocx?y* - 3x4y2 = -2 D.*?y3 - 3x3y2=-2 Ex?y2 – 3x3y2 = -2

Answer 1

) ymt - 60 scht Amo (2x-'92-94) dx + (3 7-63) dy 91-1 • Multiply the equahon by M (any) to get (uid,y) = xhym) = (2x 'y² Xx^ym 9Xyxa" ym) dx +(3yxx"ym - 6 Xxx”y m) dy = 0 =) (2xn-1 ymt .9 xhym tl) dx + (3x ym ti lym Idy=c • for this equation to be exact requires 2 (2x"*ly m+2 - 9x"ymt!) ay = 2 (33"* +) - 6xentig mo). => 2 (m+2) oc paly - 9m+1) anym 3 nachrymt! - 6(n+1)xc" ym 6ے nol mtl д 20 = (A

and xh • Now matching conreuponding terms jie, analy xhym -) 2 (m+2) = 3n and -9(m + 1) = -6 (nt) - 2m +4-35 -0 qmtq - 60-6= am-ont3 LA E 3m - 2n Elso and co B Now 3(A) -2Ⓡ - 5n +10= o 7) 3 (2m-3n+4) 2 (3m-2n +1) - 2 n= to s Emr-an+ 12-ym tun-2=0 Now 2m-3nt4=0 5) 2m - 6+4+0 =) 2m = 2 =) M=/

(A 3 2 So, the integranng fachor isbry Now put values of n and min (2 x y 3 - 9x²42) dx + (3x² y ² - 6x²y) dy = 0 Solution to this af = J (2xy 3 -9 ey 2) ax 2y? y 3 x ²-3x3y² taly) Solution to this; f = $(3*?y2-623y) ay &x²y3 'y² + B (x) -9x² y² +aly) 3 - =) 2 f(x) = 20 c²y3 - 3x sy -c at x=1 y = l =) | - 3 C D) CE-2 i. Soluhon JU 1² y 3 - 3 x3y² =-2 - Opron D