The particular solution of the differential equation (2 x-2 y+3) d x-(x-y+1) d y=0 when x=0, y=1 is

Step-by-step solution

Let x-y+1=v aligned & 1- d y ~d x = d v ~d x & d y ~d x =1- d v ~d x & now d y ~d x = 2 x-2 y+3 x-y+1 & 1- d v ~d x = 2 v+1 v & d v ~d x =1- 2 v+1 v = -(v+1) v & v v+1 ~d v=- d x & (1- 1 v+1 ) d v=- d x & n- |n+1|=-x+c & (x-y+1)- (x-y+1+1)=-x+c & 2 x-y+1+ (x-y+2)=c aligned aligned & for x=0, y=1 ; c=0 & 2 x-y- (x-y+2)+1=0 aligned