Solve dz/dx=(1+xy)^xln(1+xy)*y

$$\frac { d z } { d x } = ( 1 + x y ) ^ { x } \ln ( 1 + x y ) \cdot y$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&y\neq 0\\x>0\text{, }&x>0\text{ and }y=0\\x\in \mathrm{R}\text{, }&x=0\text{ and }y=0\\x<0\text{, }&x<0\text{ and }y=0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}\\y=0\text{, }&\text{unconditionally}\\y>0\text{, }&y>0\text{ and }x=0\\y\in \mathrm{R}\text{, }&y=0\text{ and }x=0\\y<0\text{, }&y<0\text{ and }x=0\end{matrix}\right.$$