How do I solve an electron density continuity equation for Earth's atmosphere?

[Imagin_e]

Homework Statement

Hi!

I need to solve a continuity equation for electron density as a function of time in the E-region of Earth's atmosphere. I shall neglect vertical transport and that the ion production rate completely shuts off dusk.

Homework Equations

See below

The Attempt at a Solution

Here is my attempt:

(1+λ)dNe/dt=p(t)-(1+λ)(α_d-aλ_i)Ne²-Ne dλ/dt
Ne is the electron density, dy, λ=ratio of negative ions to electrons , p(t) is the electron production rate, α_d is the recombination coeff. for ions and a_i is the recombination coeff. negative ions with positive ions.

The effective recombination coefficient is: α_eff=(1+λ)(α_d+aλ_i).
If we assume that the is no negative ions, we get:
dNe/dt=p(t)-a_dNe² (1)
and a_eff simply becomes a_d

Here comes the issue, I need to have a solution for equation (1) with respect to the density and time. . How should I integrate this? The density from the beginning is ≈ 1*10⁹ m³ , which I calculated

 

[STEMucator]

There is a typo in equation $(1)$. If I understand correctly you meant to write $\alpha_d$ not $a_d$.

Equation $(1)$ has the form of a Riccati equation:

$$\frac{dN_e(t)}{dt} = p(t) - \alpha_d N_e^2(t)$$
$$y'(t) = p(t) + g(t)y(t) + f(t)y^2(t)$$

Where $N_e(t) = y(t)$, $p(t) = p(t)$, $g(t) = 0$, and $f(t) = \alpha_d$.

To learn how to deal with equations of this form, see: https://en.wikipedia.org/wiki/Riccati_equation#Reduction_to_a_second_order_linear_equation

 

[Imagin_e]

There is a typo in equation $(1)$. If I understand correctly you meant to write $\alpha_d$ not $a_d$.

Equation $(1)$ has the form of a Riccati equation:

$$\frac{dN_e(t)}{dt} = p(t) - \alpha_d N_e^2(t)$$
$$y'(t) = p(t) + g(t)y(t) + f(t)y^2(t)$$

Where $N_e(t) = y(t)$, $p(t) = p(t)$, $g(t) = 0$, and $f(t) = \alpha_d$.

To learn how to deal with equations of this form, see: https://en.wikipedia.org/wiki/Riccati_equation#Reduction_to_a_second_order_linear_equation

THANK YOU THANK YOU THANK YOU!