Question #302636

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

1
Expert's answer
2022-02-27T16:49:26-0500

y=2x+2y+32x+2y1y=2x+2y1+42x+2y1y=1+42x+2y12x+2y1=z(x)y=12(z2x+1)y=12(z2)12(z2)=1+4z12z=2z+4zzdzz+2=4dxzdzz+2=4dxz+22dzz+2=4dx(12z+2)dz=4x+Cz2lnz+2=4x+C2x+2y12ln2x+2y+1=4x+C2x+2y12ln2x+2y+1=Cy'=\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


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