Question

Q4/ Solve the following ordinary differential equation: (ex+y + y ey) dx + (xey - 1 )dy = 0

Answer 1

Solution We have to solve the diffrential equation (exty tyes) det + (no 1) dy=0 its in the form for of hare+Ndy=0 2) Me exty + yey N= xeyl Thus am Ma 2 enty yesteg y Jy N = 2N ey dx My & Ne We Can Thus Our Differential equation is not enact. For making this find integrating factor By using the for mula dm - My-Nr. M de du M N My-Na

Here when We formala The IS We get different é y Use this integration difficult. also we how can use duftere or integrating factors. let us take integrationg factor as Thw medtipplying witte biteventual equation c/ + 4 + + (x - 2 w = a exty yell) de tes (ne 2%-1 M = 0 Now our Minyl ¿Y/ente fye) (nes-1) l N (ny? and a m(ny) ay = 2 (erty e tyes, it dy dy

and 2 N(x,y) Pyar deh (ned-1) an an Z an Our the 2) 2 miny? 2N(ny) ay Ditterential equation become exact. Thus general Solution is Smarts (terms in N not involving x) dy = Seeth nedleg.de +/-eg. dy=c 2) Sleety) de +6 - Gdy = C en tya + BE1%C 2)

z) e+Gr + el ac Thus the general Solution is given by en + Yete G=C