The solution of the differential equation d y ~d x =(x-y)^2 when y(1)=1 is

Step-by-step solution

Given differential equation is d y ~d x =(x-y)^2 Let x-y= t aligned & 1- d y ~d x = dt ~d x & ~d y ~d x =1- dt ~d x aligned From (i) aligned & 1- dt ~d x = t ^2 & 1- t ^2= dt dx aligned d x= 1 1- t ^2 dt Integrating on both sides, we get aligned & d x= 1 1- t ^2 dt & x= 1 2 | 1+ t 1- t |+ c & x= 1 2 | 1+x-y 1-(x-y) |+ c & x= 1 2 | 1+x-y 1-x+y |+ c aligned But y(1)=1 aligned & c=1 & x= 1 2 | 1+x-y 1-x+y |+1 & x-1= 1 2 | 1+x-y 1-x+y | & 2(x-1)= | 1+x-y 1-x+y | & - | 1-x+y 1+x-y |=2(x-1) aligned