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in other words, prove why the method of multiplying by mu for linear equations do not work for nonlinear ones.
Well, the idea behind multiplying by mu in a linear equation is to create that product rule derivative, much in the manner that you suggested to rewrite as (xy)'. But with non-linear, that mu may or may not exist and if it does, it certainly wouldnt be found in the same manner. I can't prove that, but the same method doesnt work because you wouldnt be able to find a guaranteed/plausible method for finding that mu, if it exists.
As for your suggestion of rewriting and then integrating both sides with respect to x, that fails because you wouldn't have a proper separation of variables. Keep in mind that in the linear case, you always have a function of x alone on the other side of the equality, but in this case you do not. The ideas behind the normal linear case just simply don't apply here. That may not be quite the answer you're looking for, but thats how I see it.