Given the differential equation (12+5xy)dx +(6xy^-1+3x^2) dy=0 which is not exac

Question

Given the differential equation (12+5xy)dx +(6xy^-1+3x^2) dy=0 which is not exact, find an integrating factor (in the form ((x^n)(y^m) ) and use it to find the general (nonzero) solution I cant come up with the integrating factor, any help would be great
Given the differential equation (12+5xy)dx +(6xy^-1+3x^2) dy=0 which is not exact, find an integrating factor (in the form ((x^n)(y^m) ) and use it to find the general (nonzero) solution I cant come up with the integrating factor, any help would be great

Explanation / Answer

non exact differential equation Question Details Given the differential equation (12+5xy)dx +(6xy^-1+3x^2) dy=0 which is not exact, find an integrating factor (in the form ((x^n)(y^m) ) and use it to find the general (nonzero) solution I cant come up with the integrating factor, any help would be great (12+5xy)dx +(6xy^-1+3x^2) dy=0..............................1 OK I SHALL SHOW YOU HOW TO FIND THE I.F. HOPE YOU CAN DO THE REST AS I UNDERSTAND FROM YOUR POST .. LET [X^N][Y^M] BE THE I.F. MULTIPLYING EQN. 1 WITH I.F.WE GET [12(X^N)(Y^M)+5{X^(N+1)}{Y^(M+1)}]dx +[6{X^(N+1)}{Y^(M-1)}+3{X^(N+2)}{Y^M}] dy=0 IF THIS IS EXACT THEN .... UDX+VDY=0 DU/DY=DV/DX SO WE GET [12M(X^N){Y^(M-1)}+5(M+1){X^(N+1)}{Y^(M)}] = [6(N+1){X^(N)}{Y^(M-1)}+3(N+2){X^(N+1)}{Y^M}] EQUATING THE RESPECTIVE POWER TERMS ON EITHER SIDE WE GET 12M=6[N+1]......................2M=N+1...............................2 5[M+1]=3[N+2]...................5M=3N+1...............................3 3*EQN.2 - EQN. 3 GIVES M=2 N=3 HENCE THE INTEGRATING FACTOR IS [X^3][Y^2]

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