यदि `y=(x)/(x+5)` तब सिद्ध कीजिएः की `x(dy)/(dx)=y(1-y)`

Explanation

`y=(x)/(x+5)`
`Rightarrow (dy)/(dx)=((x+5)(d)/(dx)x-x(d)/(dx)(x+5))/((x+5)^(2))`
`=((x+5).1-x(1+0))/((x+5)^(2))=(5)/(x+5)^(2)`
`therefore x(dy)/(dx)=(5x)/((x+5)^(2))`
`therefore xdy/dx=(5x)/(x+5)^(2)`.......(i) पुनः `1-y=1-(x)/(x+5)=(x+5+x)/(x+5)`
या `1-y=(5)/(x+5)`
`therefore y(1-y)=((x)/(x+5))((5)/(x+5))`
`Rightarrow y(1-y)=(5x)/((x+5)^(2))`.......(ii)
समीकरण (i) व (ii) से
`x(dy)/(dx)=y(1-y)`