Question

Find an integrating function of only one variable and solve the given equation

cos x cos y dx + (sin x cos y - sin xsin y + y) dy = 0

Answer 1

Am 2po cinen that. cosx cosy dat (sinecosy-sinxging tydyzo The given differential equation has the form M(49) dx + N(Xgy) dyzo Here M(x,y) = cosx cosy e N(X,Y)= Sinx cosy - sinxsing ty Now, we will find entegrating factor Moby [ SN - SMJ = kig) is a function of y only, then rutegrating facts is s kly) dy . Nowa SN = - cibx siny + cosx cosy SM -Lose siny S4 Now, I r-cosxsing +Losx cosy - (- cosasiny)] cosx cosy L isxsinyt cosx cosy tcosx si kiy) a cosxcosy k(y) = cost cosy - 1 (extef9 *19) = 1 I dyg. I of = e

Now , multiply iny by ritegrating factor et given differential equation , will get Cos x cosy e due + (siue cosy - Sewa sinyty jedyo Ma cosa co N = (sinx cosy - siux sing ty) e Now, we weate the given differential of a exact SM - Cosul ey cosy + eye-siny)] sy = cosxe cosy cosxsinyy. SN = el cost cosy - cosx siny] SN = el cost cosy - e cosx siny TSM = el cosx cost-e cose sing = d.

Now a well will find the general Solution Here M(x,y) = cosa cosy es and N(14,4)= (siux cosy - siue sing +4) et First, Integrating M(x,y) Worty we get fm (44) sx = cosa cosyes sx Son (344) 5x = siux cosyet qui Now Integrating N(44) wort y q ule get .. IN (44) sy = 1 (sinx cosy - siux siny ty je sy IN 184) by a sinx cosy e dy I sinresiuy edyt / yetdy Ja may ) le sing secesy) sine - (casy t display Jaime tely et 5) elany fiux t el cosyair + elecosy siune - Stany ghat tely el t Now el cosy seux tey- e . - quiero merge 40 aliis , write down each term exactly ance.

- Here, expression contain the terous e siux cosy gely get so that (144) = e sinx cosy tey- The general solu of the D.E es. f(x,y)=C I e sinx cosy tély -e =( This i the solution of the given D.E.