# Solving a system of ODE

Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$

$$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ where $$A,B,\eta_k$$ some known constant. $$e_k, k=1,2,3$$ are some functions we do not know the exact form, but we know the values of $$e_k$$ at the discretization points.

The thing bothers me is the right boundary condition for $$C_2,C_3$$ is the Neumann type.

My question is is there any references for solving this kind of equations? Is there any way I can convert this to a Dirichlet boundary value condition problem? I am thinking about using a shooting type method to convert this into a Dirichlet boundary condition problem, but I do not know if this will work. I could not find any references for this kind of equations.

Any help is appreciated! Many thanks!

## 1 Answer

Assuming $$\eta \ne 0$$, the general solution of $$\eta y'' = e(x)$$ is $$y(x) = c_1 + c_2 x + \eta^{-1} \int_0^x (x-t) e(t)\; dt$$ Substitute in the boundary conditions and solve for $$c_1$$ and $$c_2$$. Of course the result will depend on some integrals involving your unspecified function $$e$$.

• Thanks for your help! Could you help provide some references I can look at to learn how to get the general solution? Thanks again! – Tony Mar 19 at 16:53