Question

(15 points) A Bernoulli differential equation is one of the form dy dar + P(x)y= Q(x)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du + (1 - n)P(x) = (1 - nQ(x). dx Use an appropriate substitution to solve the equation xy' +y=2xy? and find the solution that satisfies y(1) = 1.

Answer 1

ท Given Joli ng ty=zny Dividing by any ür y! + ay 2 o res -ý=2 04 wirit est dy वर = dz an 0 =) = 2 du t in z. P= Integrating factor in dz (-1/2 ) z Thiń first order linear eah in comparing ☺ with 12 y +PZ=0, we hore -'a, B= 2 factor in da -e xl Multiplying both siden of and then integrating =/ en da [ By formula, ZxI.F. =/ I. F. X & de Jp de e th I.fi -lun ensit - е e 드 - by I. F we get Z x xt-

근 =) 7 2 ln (ng to [e in a.c) = ) 2 ln (n.) te -tay 3 -- t: z=-- Giren yll) = 1 i.l, y=1 at n=1. - 1 2 ln (lote [ln (1)=0 ༡ = -1c - hy 2 lncryl =) =) ast a zn. energ 1 yu) Y(x) = 2-2xln (2) Area)