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For Initial Value problems I want to implement an ODE solver for

Have any of you worked with this topic and can help me find the right formulas to implement? I am mostly confused about

( I have looked at this article but it does not really help me find what I am looking for https://www.newcastle.edu.au/__data/assets/pdf_file/0020/22565/50_Adaptive-backward-Euler-time-stepping-with-truncation-error-control-for-numerical-modelling-of-unsatuarted-fluid-flow.pdf )

**implicit Euler method with adaptive time step**and use**step doubling**to estimate error. I have found some reading stuff about*adaptive time step and error estimation using step doubling*but those are mostly related to RK methods. I can't find how the error estimating formulas look like for the implicit (Backward Euler) method.Have any of you worked with this topic and can help me find the right formulas to implement? I am mostly confused about

*how to update the size timestep each at each step.*( I have looked at this article but it does not really help me find what I am looking for https://www.newcastle.edu.au/__data/assets/pdf_file/0020/22565/50_Adaptive-backward-Euler-time-stepping-with-truncation-error-control-for-numerical-modelling-of-unsatuarted-fluid-flow.pdf )

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