Nonhomogeneous Linear Second-Order Differential Equations

The nonhomogeneous linear second-order differential equations with constant coefficients can be written in general form as... 

Remember 2.2 The general solution to nonhomogeneous linear second-order differential equations with constant coefficients can be obtained as the product of function to be determined and the solution to the homogeneous equation (see equation (2.41)). 

This is a rather nasty problem to solve numerically, because boundary conditions over the whole range of x are assigned at t = 0 and t = 1. A perturbation expansion around l/P = 0 yields, as expected, the CSTR at the zero-order level at all higher orders, one has a nested series of second-order linear nonhomogeneous differential equations that can be solved analytically if the lower order solution is available. The whole problem thus reduces to the solution of Eq. (133), which has been discussed before. This is, of course, the high-diffusivity limit that corresponds to a small Thiele modulus in the porous catalyst problem. 

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