The number of integral solutions of the equation $$7\left(y+\frac{1}{y}\right)-2\left(y^{2}+\frac{1}{y^{2}}\right)=9$$ is

Lets assume $$\left(y+\dfrac{\ 1}{y}\right)\ =\ t$$.

So, $$\left(y^{2}+\dfrac{1}{y^{2}}\right)$$ = $$t^2-2$$

So, the given equation can be written as 7(t) - 2($$t^2-2$$) = 9
    7(t) - 2$$t^2$$ = 5 
    Hence, 't' values could be {1} or {5/2}.

Now, $$\left(y+\dfrac{\ 1}{y}\right)\ =\ 1$$ , has no integral solution .
But for $$\left(y+\dfrac{\ 1}{y}\right)\ =\ 5/2$$, we can get y = 2 and y = 1/2.

Therefore, the only integral values of Y possible is {2} . Hence no of integral solutions will be one.