sol:- given- dy/dx=x+y+1/2x+2y+1 put x+y=v ..now diff it wrt x we get- 1+dy/dx=dv/dx => dy/dx=(dv/dx-1) put this value of dy/dx in given eq.... dv/dx=(v+1/2v+1)-1 or dv/dx=-v/2v+1 or dx=-(2v+1/v)dv or dx=-(2+1/v)dv now integrating this eq we get---- x+c=-2v-logv now put the value of v: v=x+y we get--- x+c=-2(x+y)-log(x+y) and solving this eq we get the solution as--- 3x+2y+log(x+y)+c=0_________________ required solution