# solve differential equations coupled with the finite difference method

I have these three differential equations in which I need to solve numerically:

$$\frac{dn_0}{dt}= -n_0(t)W_{01}(t) + n_1(t)K_{10}$$

$$\frac{dn_1}{dt}= -n_1(t)W_{12}(t) - n_1(t)K_{10} + n_2(t)K_{21} + n_0(t)W_{01}(t)$$

$$\frac{dn_2}{dt}= n_1(t)W_{12}(t) - n_2(t)K_{21}$$

such that

$$n_0(0)=1$$ $$n_0(N)=0$$ $$n_1(0)=0$$ $$n_1(N)=1$$ $$n_2(0)=0$$ $$n_2(N)=0$$

Using the central finite difference formula:

$$\frac{n_{0}(t + \Delta t) - n_{0}(t - \Delta t)}{2\Delta t}=-n_0(t)W_{01}(t) + n_1(t)K_{10}$$

$$\frac{n_{1}(t + \Delta t) - n_{1}(t - \Delta t)}{2\Delta t}=-n_1(t)W_{12}(t) - n_1(t)K_{10} + n_2(t)K_{21} + n_0(t)W_{01}(t)$$

$$\frac{n_{2}(t + \Delta t) - n_{2}(t - \Delta t)}{2\Delta t}=n_1(t)W_{12}(t) - n_2(t)K_{21}$$

How do I determine the functions $$n0$$, $$n1$$ and $$n2$$ knowing that $$n0 + n1 + n2 =1$$, and that the three equations are coupled?

And I could not understand how to calculate the derivatives, how can I determine their value with the finite difference method without knowing the functions?