asymptotics of a solution for equation

Question

I want to solve the equation

$$\frac{x^{17/6}}{a^{17/6}} - \frac{x^2}{a} -1 = 0$$

where I assume $a \gg1$ and $x$ is the unknown. How do I compute the dependence of $x$ on $a$ as $a \rightarrow \infty$.

Answer 1

If $x=a^{11/5}$ then $${x^{17/6}\over a^{17/6}}-{x^2\over a}=0$$ If $a$ is big, then both term in this difference are changing rapidly, so the value of $x$ leading to $${x^{17/6}\over a^{17/6}}-{x^2\over a}=1$$ will be very close to $x=a^{11/5}$. With a bit of attention to the derivative, no doubt you could get an estimate as to just how close.