Suppose we are given a system of n-1 differential equations with n-1 unknowns. $$ \begin{array}{ll} y_2 \frac{dy_2}{dt} = K_2(y_1-y_2)+K_3(y_3-y_2)\\ y_3 \frac{dy_3}{dt} = K_3(y_2-y_3)+K_4(y_4-y_3)\\ \vdots \\ y_i \frac{dy_i}{dt} = K_i(y_{i-1}-y_i) + K_{i+1}(y_{i+1}-y_i)\\ \vdots \\ y_n \frac{dy_n}{dt} = K_n(y_{n-1}-y_n) \end{array}$$ where $y_1$ and the $K_i$ are all constants. I used Wolfy-Alpha to figure out the family of solutions when n=2: $$y_2 = y_1\left(1+W\left(-\frac{1}{K_2y_1}exp\left(\frac{(c_1-t)K_2}{y_1} - 1\right)\right)\right)$$ where $W$ is the product logarithm and $c_1$ is a constant of integration.
I used a numerical iterative solution for n=3 (I can get 5 iterations into it before my computer gets cranky). The graph of the solutions to the n=3 case resembles the graph of the solution to the n=2 case in that we expect a sharp incline near t=0 and a horizontal asymptote at $y_i = y_1$ for all i (in these pictures $K_2 = 1$, $K_3 = \frac{1}{2}$ and $y_1 = 1$).
I also want to find a solution set to a similar system where, instead of the LHS, we multiply the RHS of each equation by $y_i$. Any help would be greatly appreciated. Thanks.