I How to characterise solutions to an unsolved equation

[NotEuler]

TL;DR

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)?

 

[Office_Shredder]

$g'(x)=\frac{ax+b}{cx+dx^2}$
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)

 

[NotEuler]

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.

 

[Frabjous]

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.

 

[PeroK]

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?

 

[FactChecker]

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?