Question

need help please

6. We say f(x,y) is a function of x +y if f(x,y) = g(x+y) for some one variable function g. For example, sin(a+y) and ex+w' are functions of x + y. (a) Find a condition on the differential equation A(x, y) + B(x,y) = 0 so that it may be transformed into an exact equation via an integrating factor (+ v). (b) What is a formula for this integrating factor. (c) Use this strategy to solve (7x* + 3x+y + 4y) + (4x + x + 5y)y = 0.

Answer 1

6 @ Alcay)+B606,4) '= 0 A[2C, 4) doet B (sc,y) dy=0 Af Macty) is an Integrating faction Then AM dxt BM dy=0 Then these an exact conation Then BLAM) (BM) B M # M = Mocty) then the con becomes od ulato) nad Marey) |_ A _ JB ay da doctu) dlaty) Mlxey) - 24 1 dilaty) – (244) dlarty) B-A = u(x ty

Integnating w.hd xety, we get bon plochy)=( f(c+y) (ote o IF Integnating toort tey, we get bis Mxty) = (flects) d (acty) IF = f(bc+x)= exp( sfico)(+0)) Hence Mlaty) will be Integrating faction if AaB so dx pe function of aty = f(arty) - Ane B-A ® Frien If = expl Sfente) dlare) Here Gox²+3x2y +44) dat (42²+x+54) dy=0 = 3+4,1222+1 then OB 32344 (422264) -93243 M=B-A - -32570C-Bruty - 3x2(x+y) + foto)

– е -92c4+3 d = +4) (-33241) = 8c+y Then IF erudcoces) Se rey dearey) 3log(key) (Octyla Then Ifa Octa} Trultiply there and solve