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We study differentiability properties in a particular case of the Palmer'slinearization Theorem, which states the existence of an homeomorphism $H$between the solutions of a linear ODE system having exponential dichotomy and aquasilinear system. Indeed, if the linear system is uniformly asymptoticallystable, sufficient conditions ensuring that $H$ is a $C^{2}$ preservingorientation diffeomorphism are given. As an application, we generalize aconverse result of density functions for a nonlinear system in thenonautonomous case.

Differentiability of Palmer's linearization Theorem and converse result for density functions