I'm trying to solve Hodgkin and Huxley equation given by, $$\frac{d n}{d t}=\alpha_{n}(1-n)-\beta_{n} n$$ using the boundary condition $$n_{0}=\frac{\alpha_{n_{0}}}{\alpha_{n_{0}}+\beta_{n_{0}}}$$.
The solution of this differential equation should be
$$\begin{aligned} &n=n_{\infty}-\left(n_{\infty}-n_{0}\right) \exp \left(-t / \tau_{n}\right)\\ &\text { where }\\ &\begin{aligned} n_{\infty} &=\alpha_{n} /\left(\alpha_{n}+\beta_{n}\right) \\ \tau_{n} &=1 /\left(\alpha_{n}+\beta_{n}\right) \end{aligned} \end{aligned}$$
I tried two methods.
1)Substituting $n=uv$ and equating all $v$ terms to zero. 2) Integrating factor method
But none of these methods gave me the correct answer. Can someone please suggest a method to solve this? Thank you very much!