Answer to Question #302636 in Differential Equations for Khushbu sharma

Question

Solve y^1=(2x+2y+3)/(2x+2y-1)

Answer 1

$y'=\frac{2x+2y+3}{2x+2y-1}\\ y'=\frac{2x+2y-1+4}{2x+2y-1}\\ y'=1+\frac{4}{2x+2y-1}\\ 2x+2y-1=z(x)\\ y=\frac{1}{2}(z-2x+1)\\ y'=\frac{1}{2}(z'-2)\\ \frac{1}{2}(z'-2)=1+\frac{4}{z}\\ \frac{1}{2}z'=\frac{2z+4}{z}\\ \frac{zdz}{z+2}=4dx\\ \int{\frac{zdz}{z+2}}=\int4dx\\ \frac{z+2-2dz}{z+2}=\int4dx\\ \int(1-\frac{2}{z+2})dz=4x+C\\ z-2ln|z+2|=4x+C\\ 2x+2y-1-2ln|2x+2y+1|=4x+C\\ -2x+2y-1-2ln|2x+2y+1|=C$