Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$
$$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ with $$e_1 = -\beta_1-\beta_3$$ $$e_2=-\beta_1-\beta_2$$ $$e_3=\beta_2-\beta_3$$
satisfy the constraint $$-e_k\le \frac{\eta_kC_k}{K_k+C_k}$$ where $\eta_k, K_k$ are some constant, choose $\beta_1,\beta_2,\beta_3$ such that $$19\beta_1+0.5\beta_2+16\beta_3$$ take the maximum.
Any ideas to solve this system of ODEs? I guess solving analytically is impossible, so how to solve it numerically? I have been working on this for about two weeks and I still have no clue. I found it difficult since the right hand side of the ODEs are known from optimization, so I cannot directly apply well-known methods like finite difference. Also the boundary condition contains first derivative, this makes it more difficult.
Any help is appreciated!
Thanks so much!