I How to characterise solutions to an unsolved equation

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The discussion revolves around characterizing the dependence of solutions to an equation involving an unknown differentiable function g(x). The equation presented is a complex relationship where the solution's behavior is influenced by g'(x). Participants express uncertainty about how to rigorously define this dependence, particularly when g'(x) is not constant. There is a consensus that more information about constants a, b, c, d, and the function g(x) is needed to determine specific values of x that satisfy the equation. Clarification on terminology and the original equation is also highlighted as necessary for a more precise discussion.
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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)?
 
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##g'(x)=\frac{ax+b}{cx+dx^2}##
Now integrate both sides. On the left side you get ##g(x)## plus a constant from the fundamental theorem of calculus. I'll leave it up to you to try integrating the right side (hint: try partial fraction decomposition)
 
Thank you Shredder. I may have phrased my question poorly, and am not sure if your answer really addresses what I intended to say.

What I meant is: there is some value of x that solves the original equation. That value will somehow depend on g'(x). How can that dependence be characterised?

I am not sure how integrating will help. Then I will just have an equation with g(x) on one side and a function of x on the other, and I still have the same problem.

Or perhaps I misunderstand something here.
 
NotEuler said:
Thank you Shredder. I may have phrased my question poorly, and am not sure if your answer really addresses what I intended to say.

What I meant is: there is some value of x that solves the original equation. That value will somehow depend on g'(x). How can that dependence be characterised?

I am not sure how integrating will help. Then I will just have an equation with g(x) on one side and a function of x on the other, and I still have the same problem.

Or perhaps I misunderstand something here.
Generally, the goal of these equations is to solve for the function g(x). More information is then required to get a specific value of x.
 
NotEuler said:
Or perhaps I misunderstand something here.
You ought to be precise about the whole question. Do you have "known" constants ##a, b, c,d## and a "known" function ##g(x)##? And you want to find the specific values of ##x## that solve you equation?
 
With ##x=0##, assuming that ##g'(0) \ne \infty##, we must have ##a=0##. Also, what do you mean by "original equation" in post #3?
 
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