The following ODE is obtained from the Ornstein-Uhlenbeck process. I read it from the paper Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit.

$$ \frac{\sigma^2}{2} \frac{d^2u(x)}{dx^2}+\mu(\theta-x) \frac{du(x)}{dx}=ru(x) $$

The paper does not provide the boundary conditions and I cannot find a clear explaination in its references. However, the paper just provides its solution

$$ F(x)=\int_{0}^{\infty} u^{\frac{r}{\mu}-1}e^{\sqrt{\frac{2 \mu}{\sigma^2}}(x-\theta)u-\frac{u^2}{2}}du $$

$$ G(x)=\int_{0}^{\infty} u^{\frac{r}{\mu}-1}e^{\sqrt{\frac{2 \mu}{\sigma^2}}(\theta-x)u-\frac{u^2}{2}}du $$

For this ODE, I have two questions:

(1) What are the boundary conditions to get the above two integration equations?

(2) the above two integrations are improper integraions. I have tryied to used the Matlab integral function to calculate them numerically, but for some parameters matlab raises the max interval warning. Is there routine/code can calculate these integration accurately?

Thank you.