Question

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

Answer 1

$$y'=\dfrac{2x+2y+3}{2x+2y-1}$$

Let $x+y=t.$ Then

$$t'_x=1+y'$$

$$t'-1=\dfrac{2t+3}{2t-1}$$

$$t'=\dfrac{4t+2}{2t-1}$$

$$\dfrac{2t-1}{4t+2}dt=dx$$

Integrate

$$\int \dfrac{2t-1}{4t+2}dt=\int dx$$

$$\int \dfrac{2t-1}{4t+2}dt=\dfrac{1}{2}\int \dfrac{4t+2}{4t+2}dt-\int \dfrac{1}{2t+1}dt$$

$$=\dfrac{1}{2}t-\dfrac{1}{2}\ln|2t+1|+C_1$$

$$\dfrac{1}{2}t-\dfrac{1}{2}\ln|2t+1|=x+\dfrac{1}{2}C$$

$$x+y-\ln|2x+2y+1|=2x+C$$

$$y-x-\ln|2x+2y+1|=C$$