The particular solution of the differential equation d y d x = x+y+1 x+y-1 when x = 2 3 and y = 1 3 is

Step-by-step solution

d y d x = x+y+1 x+y-1 Put x+y=v 1+ d y d x = d v d x aligned & d v d x -1= v+1 v-1 d v d x = v+1 v-1 +1= 2 v v-1 & (v-1) d v 2 v = d x & 1 2 d V- 1 2 v d v= d x= v 2 - 1 2 |v|=x+c & x+y 2 - 1 2 |x+y|=x+c aligned We have x = 2 3 , y = 1 3 aligned & 1 2 - 1 2 |1|= 2 3 +c c= 1 2 - 2 3 = -1 6 & x+y 2 - 1 2 |x+y|=x- 1 6 aligned aligned & x+y 2 - 1 2 |x+y|=x- 1 6 & (x+y)- |x+y|=2 x- 2 6 y-x+ 1 3 = |x+y| aligned