The general solution of the differential equation d y d x = x+2 y-1 x+2 y+1 is

Step-by-step solution

d y d x = x+2 y-1 x+2 y+1 Put x+2 y=t x+2 y=t 1+2 d y d x = d t d x d y d x = ( d t d x -1 ) 2 aligned & ( d t d x -1 ) 2 = t-1 t+1 d t d x -1= 2 t-2 t+1 & d t d x = 2 t-2 t+1 +1= 3 t-1 t+1 aligned aligned & t+1 3 t-1 d t= d x & 1 3 3(t+1) 3 t-1 d t= d x 1 3 3 t-1+4 3 t-1 d t= d x & 1 3 d t+ 4 3 d t 3 t-1 = d x & t 3 + 4 3 |3 t-1| 3 =x+c₁ & x+2 y 3 + 4 3 |3(x+2 y)-1| 3 =x+c₁ & 3 x+6 y+4 |3 x+6 y-1|=9 x+9 c₁ & 6(-x+y)+4 |3 x+6 y-1|=K aligned