# How to characterise solutions to an unsolved equation

• I
• NotEuler
In summary, the conversation discusses the dependence of solutions on a differentiable function g(x) in an equation with known constants. The dependence can be characterized by integrating both sides of the equation and solving for g(x), but more information is needed to get a specific value for x.

#### NotEuler

TL;DR Summary
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)?

##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?