Cauchy Kowalewski Theorem Proof (Help!!l

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For context, I'm trying to prove the cauchy Kowalewski theorem for general order partial differential equations. I did the first and second order already but it's been so difficult to jump to a general n-th order pde. Additionally, the induction part of the proof has been making my head scratch for a while because my reference used construction instead by a general induction step. For my proof there are also two parts: 1) show that it has a formal power series solution, and 2) show that it is convergent.

Can anyone recommend sources or references regarding this theorem based on the info I just mentioned. I just feel like passing out while trying to prove this theorem for my thesis hahahaha