Is this equation solvable?? $$\dfrac12= 3ax^2\cos\frac{1}{x}+ax\sin\frac{1}{x}+a^{\frac{1}{2}}$$
Solving for $a$ as a real constant without any $x$ terms. How is it possible to see if it is solvable?
 A: Wolfram Mathematica claims it cannot solve
$$\frac{1}{2}=3ax^2\cos\left(\frac{1}{x}\right)+ax\sin\left(\frac{1}{x}\right)+\sqrt{a}$$
for $x$ with the methods available to Solve, which pretty much means you really cannot express the solution in terms of standard mathematical functions.

A: If $a$ is a given constant, the equation can be solved numerically.
If $a$ is a non-given constant, you can solve the equation for $a$. You get $a$ in dependence of $x$. Take a given place $x=x_0$ and you get the constants $a$ for which the equation is fulfilled at $x=x_0$.
If $a$ is a non-given parameter, you can ask if the equation is solvable by applying only Elementary functions. An Elementary function is (in line with J. Liouville and J. F. Ritt) a function of one complex variable that can be represented as composition of a finite number of only exp, ln and/or algebraic functions. 
Subtract $\frac{1}{2}$ on each side of your equation. The left side of the equation is then a function term of an Elementary function. Let's call this function $F$. Your equation is the zeroing equation of $F$. The equation is solvable by applying only Elementary functions if the Elementary function $F$ has an inverse function that is also an Elementary function.
Convert the Elementary Standard functions in your equation (sin, cos) to their exp-ln form and you get another representation of $F$ ($i$ is the imaginary unit):
$\frac{3}{2}ax^2e^{\frac{-i}{x}}+\frac{3}{2}ax^2e^{\frac{i}{x}}+\frac{1}{2}aixe^{\frac{-i}{x}}-\frac{1}{2}aixe^{\frac{i}{x}}+\sqrt{a}-\frac{1}{2}=0$.
According to the theorem in [Ritt 1925], each Elementary function that has an elementary inverse can be represented as composition of a finite number of only exp, ln and/or unary algebraic functions. But the "monomials" of the equation, $xe^{\frac{i}{x}}$ and $x^2e^{\frac{i}{x}}$, are algebraically independent and further one can see that $F$ cannot be brought into the form of Ritt's theorem. Therefore $F$ cannot have an elementary inverse. And therefore your equation cannot be tranformed by applying only Elementary functions to yield $x$.
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90
Another approach is in
[Rosenlicht 1969] Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
If $a$ is a non-given parameter, you can also ask if the equation is solvable by other closed-form functions. Unfortunately, the "monomials" of $F$ are such that $F$ cannot be an Elementary function of Lambert W function.
