The probability density function of the ratio of two normal R.V.s
I imagine you could use the following theorem:
Let $X_1$ and $X_2$ be jointly continuous r.v.'s with joint density $f_{X_1,X_2}$. Let $Y_1=g(X_1,X_2)$ and $Y_2=g_2(X_1,X_2)$.
If
1) $y_1=g(x_1,x_2)$ and $y_2=g_2(x_1,x_2)$ can be uniquely solved for $x_1$ and $x_2$ by, say, $x_1=h_1(y_1,y_2)$ and $x_2=h_2(y_1,y_2)$
and
2) $g_1$ and $g_2$ have continuous partials with $$ J(x_1,x_2) ={\partial g_1\over\partial x_1}{\partial g_2\over\partial x_2} - {\partial g_1\over\partial x_2}{\partial g_2\over\partial x_1} \ne0, $$ then
$$ f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2} (x_1,x_2) |J(x_1,x_2)|^{-1}, $$ where
and $x_1=h_1(y_1,y_2)$ and $x_2=h_2(y_1,y_2)$.
Use the above to find the joint distribution of $Y_1=X$ and $Y_2=X/Y$ (with $X_1=X$ and $X_2=Y$); then integrate to obtain the distribution of $Y_2$.
This, by the way, is a problem in Sheldon Ross' ``A first Course in Probability'' (Chapter 6, Theoretical Exercise 33.)
Edit: This may be a ``sledgehammer''. Refer to the post suggested by Sasha...