First: cutoff, resonance, filter slope, etc., can be built with FIR (finite impulse response) filters. You can describe the passive circuit with a transfer function, which can be implemented in DSP using a z-transform. You can also describe a filter with poles and zeroes, and transform those. There's a bunch of resources on the web, e.g. https://ccrma.stanford.edu/~jos/filters/, but many assume you know something of electronics.
Recurrent filters don't have closed form solutions, and their implementation is a bit of an art. Plugin maker Urs Heckmann (from u-he.com) did a (rather successful) attempt at modeling the filter from an old synth with various implementations, and let people test them in the wild. He blogged about it, but I can't find all relevant posts. This might be a starting point: https://urs.silvrback.com/zero-delay-feedback
This course (https://www.staff.ncl.ac.uk/oliver.hinton/eee305/) is based on a book and links to the chapters, but there's a lot more (check the websites of MIT or Stanford's EE department for online courses).
Recurrent filters don't have closed form solutions, and their implementation is a bit of an art. Plugin maker Urs Heckmann (from u-he.com) did a (rather successful) attempt at modeling the filter from an old synth with various implementations, and let people test them in the wild. He blogged about it, but I can't find all relevant posts. This might be a starting point: https://urs.silvrback.com/zero-delay-feedback
This course (https://www.staff.ncl.ac.uk/oliver.hinton/eee305/) is based on a book and links to the chapters, but there's a lot more (check the websites of MIT or Stanford's EE department for online courses).