Final answer to the problem
Step-by-step Solution
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Apply the formula: $\frac{d}{dx}\left(a=b\right)$$=\frac{d}{dx}\left(a\right)=\frac{d}{dx}\left(b\right)$, where $a=\ln\left(xy\right)$ and $b=e^{xy}$
Apply the formula: $\frac{d}{dx}\left(\ln\left(x\right)\right)$$=\frac{1}{x}\frac{d}{dx}\left(x\right)$
Apply the formula: $\frac{d}{dx}\left(ab\right)$$=\frac{d}{dx}\left(a\right)b+a\frac{d}{dx}\left(b\right)$, where $d/dx=\frac{d}{dx}$, $ab=xy$, $a=x$, $b=y$ and $d/dx?ab=\frac{d}{dx}\left(xy\right)$
Apply the formula: $\frac{d}{dx}\left(x\right)$$=1$
Apply the formula: $\frac{d}{dx}\left(e^x\right)$$=e^x\frac{d}{dx}\left(x\right)$, where $x=xy$
Apply the formula: $\frac{d}{dx}\left(ab\right)$$=\frac{d}{dx}\left(a\right)b+a\frac{d}{dx}\left(b\right)$, where $d/dx=\frac{d}{dx}$, $ab=xy$, $a=x$, $b=y$ and $d/dx?ab=\frac{d}{dx}\left(xy\right)$
Apply the formula: $\frac{d}{dx}\left(x\right)$$=1$, where $x=y$
Apply the formula: $a\frac{b}{c}=f$$\to ab=fc$, where $a=y+xy^{\prime}$, $b=1$, $c=xy$ and $f=e^{xy}\left(y+xy^{\prime}\right)$
Group the terms of the equation by moving the terms that have the variable $y^{\prime}$ to the left side, and those that do not have it to the right side
Move everything to the left hand side of the equation
Apply the formula: $a\left(b+c\right)+b+c$$=\left(b+c\right)\left(a+1\right)$, where $a=-e^{xy}xy$, $b=xy^{\prime}$, $c=y$ and $b+c=y+xy^{\prime}$
Break the equation in $2$ factors and set each factor equal to zero, to obtain simpler equations
Solve the equation ($1$)
Apply the formula: $x+a=b$$\to x=b-a$, where $a=y$, $b=0$, $x+a=b=xy^{\prime}+y=0$, $x=xy^{\prime}$ and $x+a=xy^{\prime}+y$
Apply the formula: $ax=b$$\to x=\frac{b}{a}$, where $a=x$, $b=-y$ and $x=y^{\prime}$
Solve the equation ($2$)
This equation $-e^{xy}xy+1=0$ has no solutions in the real plane
The solution of the equation is
