2. Solve the equation $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$.
Answer: $x=5 / 4,5 / 3$.

Let $y=\frac{x}{\sqrt{x^{2}-1}}$, then $x^{2} y^{2}=x^{2}+y^{2}$ and $x y>0$. According to the condition, $x+y=\frac{35}{12}$,

therefore, $x^{2}+y^{2}+2 x y=\left(\frac{35}{12}\right)^{2}$, i.e., $x^{2} y^{2}+2 x y+1=\left(\frac{35}{12}\right)^{2}+1$.

Solving this quadratic equation and considering that $x y>0$, we get $x y=\frac{25}{12}$.

Now $x$ and $y$ can be easily found, knowing their sum and product.

Another way to solve it is to make the substitution $x=\sec t$.