Solve `(dy)/(dx) sqrt(1+x+y) =x+y-1`

Explanation

Putting `sqrt(1+x+y)=v`, we have
`x+y-1=v^(2)-2`
or `1+(dy)/(dx)=2v(dv)/(dx)`
Then, the given transforms to
`(2v(dv)/(dx)-1)v=v^(2)-2`
or `(dv)/(dx)=(v^(2)+v-2)/(2v^(2))`
or `int(2v^(2))/(v^(2)+v-2)dv=intdx`
or `2int[1+1/(3(v-1))-4/(3(v+2))]dv=intdx`
or `2[v+1/3log|v-1|-4/3log|v+2|]=x+c`
where `v=sqrt(1+x+y)`