How can I find the solution of the distributional equation
$$ \frac{d^2u}{dx^2}-xu = 0 $$
And prove that there are at least two linear independent solutions? I've tried to to with the Fourier Transform but it is not possible to find the solution that diverges as $x$ tends to infinity, because the Fourier Transform is not defined for such functions. (they are so called Airy functions). I've tried also to solve it by testing the equation with an appropriate test function, but I'm stuck. 
This is the method used. Once I've found Ai(x), I cannot prove that there are two independent solutions.
NOTE : The original question has been modified and completed after discussion. As a consequence my answer below is no longer well-adapted.
This ODE is known as Airy equation : http://mathworld.wolfram.com/AiryDifferentialEquation.html
There is no closed form for the solutions with a finite number of elementary functions.
A closed form requires special functions, namely the Airy functions Ai$(x)$ and Bi$(x)$ $$u(x)=c_1\text{Ai}(x)+c_2\text{Bi}(x)$$
The Airy functions are related to some other special functions, especially the Bessel functions of first kind.
To express the solutions with elementary fonctions only, infinite series are required. The calculus is rather boring. See Eqs.$(2-26)$ in the above reference.
About properties of Airy functions : http://mathworld.wolfram.com/AiryFunctions.html