Solving a system of differential-algebraic equations

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I am seeking for references around the topic of solving this type of differential equation system :

$$ \left\{ \begin{array}{ll} \partial_x y_{i+1}(x) = y_{i+1}(x)-y_{i}(x) \\ y_{i+1}(x) = y_i(f(x)) \end{array} \right. $$

where $f$ is not affine.