Numerical Ordinary Differential Equations - Boundary Value Problems

Consider a second-order linear 2-point boundary value problem (BVP) −z + p(x)z + q(x)z = r(x) (10.1) z(a) = α (10.2) z(b) = β (10.3) where p(x), q(x) and r(x) are given. By defining y(x) := [z(x), z (x)] T , the problem can be changed into a first-order differential system y = 0 1 q(x) p(x) y + 0 −r(x) (10.4) y 1 (a) − α = 0 (10.5) y 2 (b) − β = 0. (10.6) Remark. In general, a linear 2-point BVP can be written as y = A(x)y + Φ(x) (10.7) g(y(a), y(b)) = 0 (10.8) where y, Φ and g are n-dimensional vectors and A(x) ∈ R n×n. Consider the following IVP associated with (10.4), u = 0 1 q(x) p(x) u + 0 −r(x) (10.9) u(a) = α s .