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- Equation a+bx=(cx+dx^2)*g'(x). I don't know g(x) but it is differentiable. What can I say about the dependence of solutions on g(x) and/or g'(x)?

I'm pondering a seemingly simple problem: Say I have an equation with an unknown function in it. For example,

a+bx=(cx+dx^2)*g'(x)

I don't know g(x) but it is differentiable. What can I say about the dependence of solutions on g(x)?

I don't know the function g(x), except that it is differentiable.

If g'(x) is constant, this seems straightforward. What if g'(x) is not a constant? What can I say with certainty and rigor about the dependence of the solution on g'(x) and g(x)?

a+bx=(cx+dx^2)*g'(x)

I don't know g(x) but it is differentiable. What can I say about the dependence of solutions on g(x)?

I don't know the function g(x), except that it is differentiable.

If g'(x) is constant, this seems straightforward. What if g'(x) is not a constant? What can I say with certainty and rigor about the dependence of the solution on g'(x) and g(x)?