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For ODE, it is better to use high order methods with variable step, for example, Dormand-Prince method 8, or extrapolation algorithms.

Most nonlinear solution algorithms are best suited for a certain class of problems.
Newtons method: if the hessian is easy to compute and evaluate
Quasi Newton Methods e.g Gauss Newton Algorithm: for cases where you do not have an hessian, you pay with convergence
conjugate gradient methods: good for large sparse matrices
Trust region methods: better for quadratic like problems
For ODES
this depends on the order of error you're willing to tolerate
ODE45, Dormand Prince
or if youre dealing with a stiff differential equation then you need ODE23s or ODE15s in MATLAB.

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