Second ODE, initial conditions are zeros at infinity

[omyojj]

second ODE, initial conditions are zeros at infinity!

I want to know the temperature profile of phase transition layer in the interstellar medium.
For stationary solution, the dimensionless differential equation I ended up with is

$$\frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T}$$
where $f(T)$ is some complicated but well-behaved function.
Boundary conditions are
$$T(x=-\infty) = 1 ,$$
$$\frac{dT}{dx}(x=-\infty) = 0,$$

However, $f(T=1) = 1$, one obtains $\frac{d^2T}{dx^2}(x=-\infty) = 0$

How do I solve this numerically? where should I start the integration? and what should be the initial condition?
Do I need to Taylor expand the differential equation?

Thank you for your attention.

 

[MathematicalPhysicist]

One approach is to multiply by dT/dx this eqaution to get:

$1/2 d/dx (dT/dx)^2 = (f(T)/T^2-1/T)dT/dx = (f(T)-T)/T^2 dT/dx$

Now perhaps use an iterating scheme of recursion equations, and integrate from minus infinity to some arbitrary x.

Hope I helped somehow.

 

[omyojj]

hmm..can you elaborate on the iterating scheme of recursion eq. or share some links I can refer to?