Question

(10) 2. Solve the homogeneous equation by making the substitution y = xv y' x + 2y 2x + y' > 0.

Answer 1

do polve homageneous equation by making substite ton Y=vn. y = anty x+2y 7 2+24 dy da anty Given that xv equation with repect to differentialing abuse dy vt ndr da da put value of and value of y da equation (1) 70+ 271 vt adr da vtadu dre xdu da x du 2ut av *(1+2v) (2+v) 1+2v V 2+1 1+2V (v (240) 2+1 1+ 24 - 2 -V? 2 tv 1-V? 2 tv dn x du dn ndu da dr V+ 2 dv = 1-V? 7. Integrating hoth Sideg vtedra 1-1² The da he

2 dv 6) V dv t 1-v2 da 71. 1-V? T I IT Now we 1-V dv ln / It will solve all in leqal.fon 1. His put 1-4²=t -2y = dt -dt v dv. 2 dt I lnct 2 t 2 - 1 dn (1-12). Now solving second interuali 1 du Now solving Merbal 2 = 2 1 1-V 1 integral (2). putting all values in In (a + c. 10 (9-vol + in va 2 JV 112 2 (3-12] t Instv unlml+c. Using property above In at lnb= ln (ab) in oquation we get,

well 13 v.] *** toote (S1-v)(3-1) un una & C. Jan Titu In la ta 312 all team are in Un} than the Songlant In Info + Gov)317 fost We knapt that ve 뽑 In Inc (g- Ylx)² 312 using propating if tha= lit, then Q=b J + 2x A- 14 Ph. ofquearning 1+ (% (1 - 773 both Wides D (²= Cles both are Considered constant]

Do the complele Solution 13. f =2 C n² 1- h V