M=(y−xy)dydM=1−xyN=xdydN=1which is not exact(dydM−dydN)=f(x)(1−x−1)=f(x),−x=f(x)N1(dydM−dydN)=f(x)x1(−x)=f(x)−1=f(x)find the integrating factorN1(dydM−dydN)=f(x),I.F=e∫f(x)dxM1(dydM−dydN)=f(y),I.F=e∫−f(y)dxI.F=e∫f(x)dx=e∫−1dx=e−xe−x[(y−xy)dx+xdy=0](ye−x−xye−x)dx+xe−xdy=0test for exactnessdydM=e−x−xe−x,dydN=e−x−xe−xdxdF=M(x,y)∫dF=∫(ye−x−xye−x)dxF=−ye−x−y∫xe−xdxwe use integration by part to solve ∫xe−xdx∫xe−xdx=xe−x+e−xF=−ye−x+yxe−x+ye−x+Q(y)F=xye−x+Q(y)dydF=N(x,y)∫dF=∫(xe−x)dyF=(xe−x)∫dyF=xye−x+R(x),thenF=xye−x+Q(y)=xye−x+R(x)Q(x)=R(x)=0xye−x+Q(y)=Fxye−x=cThe general equation
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