I am involved with a kind of differential equation but I don't know how to approach it or classify it.
$\frac{dy(x)}{dx} = f(y(x),x(s),s)$
Where $x=x(s)$
Assume that I have 2 of those equations and my aim is to solve y(x). However x also depends on a parameter s and I want to find a y(x) such that above equation is satisfied for $a<s<b$.
So, it includes combination or some kind of nested function structure. While I was searching something about such equations, I have ended up with differential equations including functionals, integral equations and some parametric ODE's but I haven't exactly figured it out.
What can be done is, assuming that $s=s_0$ or taking the s as a parameter and varying it, get different solutions for $y(x)$ but still as you may understand I am kind of lost. As an example;
$\frac{dy}{dx}=y*(x+s)$
And since we have two unknowns let be a 2nd equation;
$\frac{dx}{ds}y=\frac{dy}{dx}*cos(x+s)$
(Of course these are nonlinear ones and probably have no easy solution or one can use seperable equations approach etc. but I just want to learn how to classify and learn something about such equations)
(y and (x+s) are in multiplication y is y(x) here and x=x(s) so as s changes also x get affected and y )
Can you suggest differential equations topics related to such equations to learn about or am I missing some very easy part in there?
Your explanations are quite unstructured and unclear, but let me try to speculate. You have something like \begin{align} \frac{dy}{dx} &= f\big(y(x), x(s), s\big)\\ \frac{dx}{ds} &= g\big(y(x), x(s), s\big) \end{align} Is this what you have? Then simply $x=x(s)$ and then $y = y(x) = y(x(s)) = y(s)$, i.e. treat $y$ as a function of $s$. Then $\frac{dy}{ds} = \frac{dy}{dx} \, \frac{dx}{ds}$ and so \begin{align} \frac{dy}{ds} &= f\big(y(s), x(s), s\big)\, \frac{dx}{ds}\\ &= f\big(y(s), x(s), s\big)\, g\big(y(s), x(s), s\big)\\ \frac{dx}{ds} &= g\big(y(s), x(s), s\big) \end{align} which is the ODE system \begin{align} \frac{dy}{ds} &= f\big(y(s), x(s), s\big)\, g\big(y(s), x(s), s\big)\\ \frac{dx}{ds} &= g\big(y(s), x(s), s\big) \end{align} or in shorter notation \begin{align} \frac{dy}{ds} &= f\big(y, x, s\big)\, g\big(y, x, s\big)\\ \frac{dx}{ds} &= g\big(y, x, s\big) \end{align}