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Slove y(xy+1)dx+x(1+xy+x^2y^2)dy=0

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Solution:  Let   xy=v

              Rightarrow   y=v/xRightarrow dy=xdv-vdx/x^2

             Now ,  diffrential equation becomes

             Rightarrow    (v/x)(v+1)dx+x(1+v+v^2)[xdv-vdx/x^2]=0

              On solving , we get

           Rightarrow     v^3dx-x(1+v+v^2)dv=0

           Separating the variables & integrating we get

          Rightarrow       int dx/x-int [1/v^3+1/v^2+1/v]dv=0

         Rightarrow      ln x+1/2v^2+1/v-ln v=c

        Rightarrow       2v^2ln (v/x)-2v-1=-2cv^2Rightarrow 2x^2y^2ln(xy)-2xy-1=kx^2y^2 

Posted by

Deependra Verma

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