LECTURE 10 Integrating Facts Recall that a differential equation of the fm (101) M(x,y)+N(x,y)y =0 issaidtobeexactif (102) = andthatinsuchacase,wecouldalwaysfindanimplicitsolutionofthefm (103) ψ(x,y)=c with (104) = M(x,y) = N(x,y) Evenif(101)isnotexact,itissometimespossiblemultiplyitbyanotherfunctionofxand/ytoobtain anequivalentequationwhichisexact Thatis,onecansometimesfindafunctionµ(x,y)suchthat (105) µ(x,y)m(x,y)+µ(x,y)n(x,y)y =0 isexact Suchafunctionµ(x,y)iscalledanntegratingfact Ifanintegratingfactcanbefound,then the iginal differential equation (101) can be solved by simply constructing a solution to the equivalent exact differential equation(105) E 101 Considerthedifferentialequation This equation is not exact; f (106) andso x 2 y 3 +x ( 1+y 2) dy =0 = = ( x 2 y 3) = 3x ( ( )) 2 x 1+y 2 = 1+y 2 However, if we multiply both sides of the differential equation by µ(x,y)= 1 xy 3 weget x+ 1+y2 dy y 3 =0 which is not only exact, it is also separable The general solution is thus obtained by calculating, H 1 (x) = x= 1 2 x2 H 2 (y) = 1+y 2 y 3 dy= 1 2y 2 +ln y 35
10 INTEGRATING FACTORS 36 andthendemandingthatyisrelatedtoxby H 1 (x)+h 2 (y)=c 1 2 x2 1 2y 2 +ln y =C Now, in general, the problem of finding an integrating fact µ(x, y) f a given differential equation is very difficult In certain cases, it is rather easy to find an integrating fact 01 Equations with Integrating Facts that depend only on x Consider a general first der differential equation (107) M(x,y)+N(x,y) dy =0 We shall suppose that there exists an integrating fact f this equation that depends only on x: (108) µ=µ(x) Ifµisteallybeanintegratingfact,then (109) µ(x)m(x,y)+µ(x)n(x,y) dy must be exact; ie, (1010) (µ(x)m(x,y))= (µ(x)n(x,y)) Carryingoutthedifferentiations(usingtheproductrule,andthefactthatµ(x)dependsonlyonx),weget (1011) µ = N+µ = 1 ( ) N µ Now if µ is depends only on x (and not on y), then necessarily depends only on x Thus, the selfconsistency of equations (108) and (1011) requires the right hand side of (1011) to be a function of x alone Wepresumethistobethecaseandset p(x)= 1 ( ) N sothatwecanrewrite(1011)as (1012) +p(x)µ=0 Thisisafirstderlineardifferentialequationfµhatwecansolve! Accdingtothefmuladeveloped in Section 21, the general solution of(1012) is [ ] (1013) µ(x) = A exp p(x) [ ( ) ] 1 =Aexp N Thefmula(1013)thusgivesusanintegratingfactf(107)solongas ( ) 1 N dependsonlyonx
10 INTEGRATING FACTORS 37 02 Equations with Integrating Facts that depend only on y Consider again the general first der differential equation (1014) M(x,y)+N(x,y) dy =0 We shall suppose that there exists an integrating fact f this equation that depends only on y: (1015) µ=µ(y) Ifµisteallybeanintegratingfact,then (1016) µ(y)m(x,y)+µ(y)n(x,y) dy must be exact; ie, (1017) (µ(y)m(x,y))= (µ(y)n(x,y)) Carryingoutthedifferentiations(usingtheproductrule,andthefactthatµ(y)dependsonlyony),weget (1018) dy M+µ =µ dy = 1 ( ) M µ Nowsinceµisdependsonlyony(andnotonx), thennecessarily dy dependsonlyony Thus,theselfconsistency of equations (1015) and (1018) requires the right hand side of (1011) to be a function of y alone Wepresumethistobethecaseandset sothatwecanrewrite(1011)as (1019) p(y)= 1 M ( ) dy +p(y)µ=0 Accding to the fmula developed in Section 21, the general solution of(1019) is [ ] [ ( ) ] 1 (1020) µ(y) = A exp p(y) =Aexp M Thefmula(1020)thusgivesusanintegratingfactf(1014)solongas ( ) 1 M dependsonlyony 03 Summary: Finding Integrating Facts Suppose that (1021) M(x,y)+N(x,y)y =0 is not exact AIf (1022) F 1 = dependsonlyonxthen ( (1023) µ(x) = exp will be an integrating fact f(1021) N ) F 1 (x)
10 INTEGRATING FACTORS 38 BIf (1024) F 2 = dependsonlyonythen ( (1025) µ(y) = exp will be an integrating fact f(1021) M ) F 2 (y)dy CIfneitherAnBistrue,thenthereislittlehopeofconstructinganintegratingfact E 102 (1026) (3x 2 y+2xy+y 3 )+(x 2 +y 2 )dy=0 Here Since this equation is not exact M(x,y) = 3x 2 y+2xy+y 3 N(x,y) = x 2 +y 2 =3x2 +2x+3y 2 2x= Weseektofindafunctionµsuchthat is exact Now F 1 F 2 µ(x,y)(3x 2 y+2xy+y 3 )+µ(x,y)(x 2 +y 2 )dy=0 = 3x2 +2x+3y 2 2x N x 2 +y 2 = 3(x2 +y 2 ) x 2 +y 2 =3 = 2x 3x2 2x 3y 2 M 3x 2 y+2xy+y 3 = 3 ( x 2 +y 2) 3x 2 y+2xy+y 3 Since F 2 depends on both x and y, we cannot construct an integrating fact depending only on y from F 2 However,sinceF 1 doesnotdependony,wecanconsistentlyconstructanintegratingfactthatisa function of x alone Applying fmula(1023) we get ( ) µ(x)=exp F 1 (x) [ =exp ] 3 =e 3x Wecannowemploythisµ(x)asanintegratingfacttoconstructageneralsolutionof e 3x (3x 2 +2x+3y 2 )+e 3x (x 2 +y 2 )y =0 which,byconstruction,mustbeexact Soweseekafunctionψsuchthat (1027) = e 3x (3x 2 y+2xy+y 3 ) = e 3x (x 2 +y 2 ) Integratingthefirstequationwithrespecttoxandthesecondequationwithrespecttoyyeilds ψ(x,y) = x 2 ye 3x + 1 3 y3 e 3x +h 1 (y) ψ(x,y) = x 2 ye 3x + 1 3 y3 e 3x +h 2 (x) Comparingtheseexpressionsfψ(x,y)weseethatwemsuttakeh 1 (y)=h 2 (x)=c, aconstant Thus, function ψ satisfying(1027) must be of the fm ψ(x,y)=e 3x x 2 y+e 3x y 3 +C Therefe, the general solution of(1020) is found by solving e 3x x 2 y+e 3x y 3 =C
fy 10 INTEGRATING FACTORS 39