i posted previously comments on y'[x]^2==y[x]^2 which i removed because of a sign error (when i woke up and rechecked it) made it unlike the other problem, it was not an envelope of the original problem to compare with.

I see carl wolf's graph doesn't change your answer (it shows both your graphs combined but doesn't answer if there is another solution line or points in another region to your 2nd diff eq). You said specifically you wished the solution of the second equation to operate using the interpolation function of the first, which the example removes. but it's a good idea to keep things together.

I plotted it and DSolve can solve it and shows slightly different graph for the first solution but the same for the second - i didn't know if that was a question.

I cannot comment if one needs to check for extraneous solution of differential equations, as I know little of what the underlying code does and does not do. I just assume one should check the answer is "correct" by hand, ask questions about particular abilities, to "know the tool" they work with.

I know NDSolve uses one of many numerical approximation methods (if it needs to), which may have limitations. I'm unsure why you would start with NDSolve then ask backwardly if the solution is extraneous or partial when DSolve shows the solution.

You didn't test all initial values, ie if y[x0]=y0, y'[x0]=y1 (noting I didn't check if your equation is linear nor if each part of the equation in the linear combination of solutions is linear).

In all: I cannot do all that at the moment. but it's safe to say you should thoroughly check the initial equation and solution and check for every possible hole. This is just expected for solving by hand.

It all depends if you are getting solutions that satisfy your need. If it is in the neighborhood and works, use it.