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My pair of coupled nonlinear differential equations

Hello every one,

In my physics problem, i end up having two coupled second-order nonlinear differential equations where the coupling terms include, the variable, the first derivatives, and also a second derivative coupling. I appreciate any help on how to handle this system before setting it for numerical simulation: that system generally looks like the following equations;

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-For my first trial, i have eliminated the term with d^2(y) from the first equation using the second equation. similarly i eliminated the term with d^2(x) from the second equation using the first. Hence i obtained a new coupled system that looks more complicated but numerically solvable. i would like to know if this process is mathematically valid or not. i.e. does the new coupled equations typically represent or give the same results as the old one.

-For my second suggestion is just to neglect those higher coupling terms from both equations then solve them, however, i'm not sure how valid that is, anything in math or physics can validate neglecting those terms.

-What is the best (standard) way to solve such system, any creative approcah as i'm not really well-informed about coupled DE?

-My final question is, any recommendation for a good book that discuss these coupled differential equations especially ones like the above.

Thanks in advance.