Constraint-Based Testing (CBT) is the process of generating test cases against a testing objective by using constraint solving techniques. In CBT, testing objectives are given under the form of properties to be satisfied by program's input/output. Whenever the program or the properties contain disjunctions or multiplications between variables, CBT faces the problem of solving non-linear constraint systems. Currently, existing CBT tools tackle this problem by exploiting a finite-domains constraint solver. But, solving a non-linear constraint system over finite domains is NP hard and CBT tools fail to handle properly most properties to be tested. In this paper, we present a CBT approach where a finite domain constraint solver is enhanced by Dynamic Linear Relaxations (DLRs). DLRs are based on linear abstractions derived during the constraint solving process. They dramatically increase the solving capabilities of the solver in the presence of non-linear constraints without compromising the completeness or soundness of the overall CBT process. We implemented DLRs within the CBT tool TAUPO that generates test data for programs written in C. The approach has been validated on difficult non-linear properties over a few (academic) C programs.