A boundary value problem "discussion" So, let's say we are given a function f : [0, 1] --> R and constants a, b, and we want to find u : [0, 1] --> R such that u''(x) + f = 0 on <0, 1> with u(1) = a and u'(0) = -b. One can easily obtain the exact solution to this problem merely by using direct integration. But, my book says: "We are interested in developing schemes for obtaining approximate solutions to this problem that will be applicable to much more complex situations in which exact solutions are not possible." Well, I'm interested in what kind of "complex situations" the author was referring to here. Is it simply the case when the given function f is not integrable? Although this post only demonstrates my lack of knowledge in analysis, I'm still curious about it, since I want to be fully motivated to start doing some finite element "research". Thanks in advance.