Regarding Julia’s speed and accuracy for ODEs, Chris compares Julia’s solvers with solvers from other languages (including Matlab) here: ODE Solver Multi-Language Wrapper Package Work-Precision Benchmarks (MATLAB, SciPy, Julia, deSolve (R))
…and a more-general comparison about the differential equation tools in several languages: A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran - Stochastic Lifestyle