Question

Solve the differential equation  `dy/dx = (x + y+2)/(2(x+y)-1)`

Answer 1

`dy/dx = ((x+y)+2)/(2(x+y)-1)`
Let  `x+y=t`

`1+ dy/dx =dt/dx  rArr dy/dx = dt/dx-1`

`dt/dx-1= (t+2)/(2t-1)`

`dt/dx =(t+2)/(2t-1)+1`

`(dt)/dx = (t+2+2t-1)/(2t-1)`

`dt/dx=(3t+1)/(2t-1)`

`((2t-1)/(3t+1))dt = dx`

`2/3 int (6t -3)/(6t+2)dt=int dx`

`2/3 int((6t+2)-5)/((6t+2)) dt = x+C_1`

`2/3 int dt - 10/3 int dt/(6t + 2)=x+C_1`

`2/3t - 5/3 xx(In|3t+ 1|)/3 = x + C`

`2/3t - 5/9 xx(In|3t+ 1|) = x + C`

`2/3(x+y)-5/9In|3(x+y)+1|=x+C`

`2/3(x+y)- 5/9 In|3x +3y +1|=x+C`