Question

please show all steps

1.)

(a) In each of (1) and (2), determine whether the given equation is linear, separable, Bernoulli, homogeneous, or none of these. (1) y = yenye (2) x²y = 3x cos(2x) + 3xy (b) Find the general solution of (1).

2.)

3.

Answer 1

• Solution: - 21(a) c) y'= xy?_4 - dy = xy! - T he Chain to en q = y3 dy+ = yº_0) compare the above equation ci) with dy + Pcx) y = Qwyn we get Pol) = 4x & 20) = 1 Ditterential equation is Bernouli's differential equcetion. ci) pey' = 3x cos(2x) + 3xy oé dy = 3x Cos (2X) + 3xy en els = 32 gos (22) + Borey = 3 cos (2x) + zby den - y = 2.cos(x) — «) compare the above equation (l) with dy + AXOY - 200) where p(x) = -3 and Qox) = 3.605(2x) : Differential equation () is Limecer differential equation 92 х

Q.1 (b) generel y = xy - y solution of (1) dy + y = y3 which is Bernozeli's differenthical equation Divide equ (1) both sides by 'y3', we get LO - Mºd + g° = 1 _8) Cussume that y = v =) -eydy - dv : from expucation (2), we get, įv = 1 multiply above equal by (2) both sides, we get, dv - LV=-2 - B) which is linecer ditterential equation in de' & 'v, compure, egh (3) with dy + Poxdy = Q(x) form, we get- P(V) = -2 and Q0W) = -2 I.F. - ej PoX) dx = 12 de = 22.00 x =elinear +१

The general Solution of eq" (3) is : V. CI.F) = f Qw (J.F) dx + c = y ² 2 2 = f - esce dx + c s keye = -2 z2t + - 9+1 olue = et ca (2+ (x) och a ł o ni x Me = 2x+cxe y² = exet coeld is which is required general sohction.