अवकल समीकरण `sqrt(1+x^(2)+y^(2)+x^(2)y^(2))+xy(dy)/(dx)=0` का व्यापक मान ज्ञात कीजिए।

Explanation

माना `sqrt(1+x^(2)+y^(2)+x^(2)y^(2))+xy(dy)/(dx)=0" "…….(1)`
`impliessqrt((1+x^(2))+y^(2)+(1+x^(2)))+xy(dy)/(dx)=0`
`sqrt((1+x^(2))(1+y^2))+xy(dy)/(dx)=0`
`xy(dy)/(dx)=-(sqrt(1+x^(2))sqrt(1+y^(2)))`
चर पृथक करने पर
`(y)/(sqrt(1+y^(2)))dy=-int(sqrt(1+x^(2)))/(x^(2)).xdx" "......(2)`
यदि `1+x^(2)=u^(2)`
`implies2xdx=2uduimpliesxdx=udu`
व `1+y^(2)=v^(2)`
`implies2ydy=2vdv`
`impliesydy=vdv`
तब यह मान समीकरण (2) में रखने पर
`int(vdv)/(v)=-int(uxxudu)/((u^(2)-1))`
`intdv=-int(u^(2))/((u^(2)-1))du=-int((u^(2)-1)+1)/((u^(2)-1))dx`
`v=-u-(1)/(2)log((u-1)/(u+1))+c`
`impliessqrt(1+y^(2))=-sqrt(1+x^(2))-(1)/(2)log[(sqrt(1+x^(2))-1)/(sqrt(1+x^(2))+1)]+c`