I am facing a problem with a numerical integration on Mathematica. I want to integrate the following system of two nonlinear 2nd order equations $$f''(x)+\lambda f'(x)=-\alpha \left(\frac{g'(x)^2}{f(x)}+2\frac{g(x)^2}{f(x)}+\frac{2 g(x)^2g'(x)}{f'(x)^2}\right)$$ and $$g''(x)-\frac{f'(x) g'(x)}{f(x)}-2 \frac{g(x) g'(x)}{f(x)}-2 g(x)=0$$
($\alpha$ and $\lambda$ are constant parameters) whose initial conditions are $f(0)=0, f'(0)=1, g(0)=1$ and $g'(0)$ is adjustable so that one must achieve the asymptotic behavior $g(\infty)\to 0$. However I am not succeeding in obtaining such expected behaviour at infinity by merely adjusting $g'(0)$. So I am wondering if there was any specific procedure to include the condition $g(\infty)\to 0$ along with the initial conditions, in order to compel such a behavior. I'd be happy with any help on this issue. Cheers.