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This paper presents an alternative approach to the deterministic global optimisation of problems with ordinary differential equations in the constraints. The algorithm uses a spatial branch-and-bound approach and a novel procedure to build convex underestimation of nonconvex problems is developed. Each nonconvex functional in the original problem is underestimated by adding a separate convex quadratic term. Two approaches are presented to compute rigorous values for the weight coefficients of the quadratic terms used to relax implicitly known state-dependent functionals. The advantages of the proposed underestimation procedure are that no new decision variables nor constraints are introduced in the relaxed problem, and that functionals with state-dependent integral terms can be directly handled. The resulting global optimisation algorithm is illustrated on several case studies which consist in parameter estimation and simple optimal control problems.