Solving this first-order differential equation for neutron abundance

[gurbir_s]

The time rate of change of neutron abundance $X_n$ is given by
$$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$
where $\lambda$ is neutron production rate per proton and $\hat\lambda$ is neutron destruction rate per neutron.
Given the values of $\lambda$ and $\hat\lambda$ at various values of time, I need to calculate $X_n$.I have also calculated values of $\lambda 's$ at intermediate times. I have tried using Euler method and RK4 method to solve this equation, but the solutions for $X_n$ diverge to inf values.

[Here][2] is the link to the complete research paper "Primordial Helium Abundance and the Primordial Fireball. II" by P.J.E. Peebles.

Any help or idea on how to solve it will be appreciated : ) [1]: Data for $\lambda 's$ https://i.stack.imgur.com/lnW9M.png
[2]: https://ui.adsabs.harvard.edu/abs/1966ApJ...146..542P/abstract

 

[BvU]

hello @gurbir_s ,

$\qquad$ !​

It seems ([edit]: (*) ) to me you have a differential equation at hand of the so-called 'very stiff' category.
I don't know what tools you have available, but you can try to find an impementation of the Gear method.

(*) the 'primordeal fireball' in the title says it all[edit2]:
A little googling: in https://globaljournals.org/GJSFR_Volume13/2-Numerical-Approach-for-Solving-Stiff.pdf
I find

12. Hindmarsh, A. C. and Gear C.W. (1974), “Ordinary differential equation system solver”, L.L.L. Report UCID -30001, rev. 3, l.l.l. (www.netlib.org/ode/epsode.f)

Good old Fortran !

$\$

 

[gurbir_s]

hello @gurbir_s ,

$\qquad$ !​

It seems ([edit]: (*) ) to me you have a differential equation at hand of the so-called 'very stiff' category.
I don't know what tools you have available, but you can try to find an impementation of the Gear method.

(*) the 'primordeal fireball' in the title says it all[edit2]:
A little googling: in https://globaljournals.org/GJSFR_Volume13/2-Numerical-Approach-for-Solving-Stiff.pdf
I find Good old Fortran !

$\$

Thank you : ) @BvU. I was struggling with this problem from quite a few days.