Differential equation of a path Differential equation of a path of a particle is $$\frac{d^2 u}{d \theta^2}-\frac{5}{4}u=\frac{9}{4}\frac{\alpha}{\beta^2}$$ where $u=\frac{1}{r}$ and $r$ is distance from origin to a particle, $\alpha,\beta=\text{const}.$   
I've got that solution of this DE is $$u=C_1e^{\sqrt{\frac{5}{4}}\theta}+C_2e^{-\sqrt{\frac{5}{4}}\theta}-\frac{9\alpha}{5\beta^2}$$
Initial conditions are $r=\frac{\beta^2}{3\alpha}$ and $\theta=0$ at moment $t=0$. Also, I know that $\frac{du}{d\theta}(0)=-\frac{3\sqrt 3\alpha}{\beta^2}$  
From these initial conditions I found constants $C_1$ and $C_2$.  
I have to show that a particle will return to its initial position after one revolution around $O$ and then fly off to infinity.  
How can I do this?
 A: The question is slightly misleading: you are not trying to match $u(2 \pi)$ with $u(0)$.  Rather, you are just trying to show that there is another $\theta$ for which $u(\theta)=u(0)$.  This is easily done.
I will skip the algebra in matching the initial conditions and produce the solution explicitly:
$$\frac{\beta^2}{\alpha} u(\theta) = \frac{3}{5} \left (4-\sqrt{15}\right) e^{\sqrt{5} \theta/2} + \frac{3}{5} \left (4+\sqrt{15}\right) e^{-\sqrt{5} \theta/2} - \frac{9}{5}$$
We wish to solve $u(\theta) = 3 \alpha/\beta^2$ for $\theta$.  As you can probably surmise, this leads to a quadratic:
$$\left (4-\sqrt{15}\right) y^2 - 8 y + \left (4+\sqrt{15}\right)=0$$
where $y=e^{\sqrt{5} \theta/2}$.  The solutions are $y=1$, for which $\theta=0$ as expected, and
$$y=\frac{4+\sqrt{15}}{4-\sqrt{15}} = \left ( 4+\sqrt{15}\right)^2$$
or
$$\theta = \frac{4}{\sqrt{5}} \log{\left ( 4+\sqrt{15}\right)} \approx 3.6932$$
So this returns to its initial point not really after a full revolution, but perhaps a little more than a quarter turn. 
As for shooting off to infinity, that is some $\theta$ for which $u(\theta)=0$.  This is another quadratic:
$$\left (4-\sqrt{15}\right) y^2 - 3 y + \left (4+\sqrt{15}\right)=0$$
Here, I get both positive solutions, and the corresponding $\theta$ occur prior to the value I found above, which doesn't fit into the narrative of the problem.
