Recently, recurrent neural networks have been extensively utilized to address a time-dependent system of linear equations (TDSLEs) with inequality systems. Nevertheless, these existing studies only limit the variable without considering constraints on its derivatives, which may be challenging to accomplish a given task in practical applications when additional constraints are introduced. Beyond that, the matrix pseudoinverse is performed, and non-negative slack variables are introduced in the solution process, which increases the model’s complexity and leads to a high computational burden. To remedy these deficiencies, this article makes improvements via proposing a novel recurrent neural dynamics (RND) model for solving the TDSLEs with constraints on the variable and its derivatives. Specifically, such a model neither needs to compute the pseudoinverse of a matrix nor to introduce non-negative slack variables, thereby enhancing its computational efficiency and accuracy. Corresponding theoretical analysis is provided to ensure its convergence performance. Finally, numerical results, comparisons with other models, and applications to single and multiple robots are provided, which substantiates the availability and meliority of the pseudoinverse-free RND model for disposing of the TDSLEs with constraints on the variable and its derivatives.