Sensitivity of ODE Solutions and Quantities of Interest with Respect to Component Functions in the Dynamics

The PDF proves exactly the same result using the same operator setup and Banach-space Implicit Function Theorem. It defines ψ(x,g) = (F2(x,g) − x′, x(t0) − x0) on X = (W^{1,∞}(I))^{n_x}, G = G2, Z = (L∞(I))^{n_x} × R^{n_x}, establishes that F2 is C^1 with derivative [fx + fg gx]·δx + fg·δg(·,x(·)) (Theorem 2.8), derives D_xψ and D_gψ (Corollary 2.9), shows D_xψ is bijective via the linear IVP δx′ = Aδx − r, δx(t0)=r0 (Lemma 2.10), then applies the IFT to obtain a C^1 map g ↦ x(g) and the sensitivity equation δx′ = Aδx + Bδg(·,x(·)), δx(t0)=0 (Theorem 2.12) . The candidate solution reproduces these steps and formulas, including the same derivative, isomorphism argument for D_xψ, and the resulting sensitivity ODE. Minor presentational differences (e.g., a contraction-mapping proof for the linear IVP and emphasizing local mapping properties of F2 near the base point) do not affect correctness.