= Solution
Integrate the <joint probability density> over $y$ to obtain the <marginal density> of $X$:
$$
f_X(x)=\int_0^\infty xe^{-x(y+1)}\,dy=e^{-x},\qquad x>0.
$$
Thus $X$ has an <exponential distribution> of rate one. Dividing the <joint probability density> by this <marginal density> gives the <conditional density>
$$
\boxed{f_{Y\mid X=x}(y)=xe^{-xy}\mathbf1_{\{y\geq0\}},\qquad x>0.}
$$
Equivalently, $Y\mid X=x$ has an <exponential distribution> of rate $x$. The value at $x=0$ can be assigned arbitrarily: it is a probability-zero conditioning value and the density ratio there is not meaningful.
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