Fore more explanation :
I am searching for an analytical solution/method to solve the equation of moving medium with space dependent coefficients. In fact, the separation variable method is used to express the unknown displacement, then the initial wave equation of a moving medium (string/cable/beam...) is reformulated such as the new unknowns are the modal shapes (since the time dependent function/modal coordinates are assumed to describe simple harmonics; an assumption generally adopted in linear dynamics as I think!!). The obtained differential equation has a space dependent coefficient (the tension/stress in the medium as well as the travelling velocity (not the wave celerity!! ) are considered as space dependent functions for sake of generality). Moreover, as it is well known, this differential equation contains complex coefficient (also space-dependent) due to the travelling velocity.
So, my question will be :
How can I solve this differential equation ? It is a differential equation with space dependent coefficients in the most general term.
P.S : I am not interested in the determination of time contributions/modal coordinates but I am searching for an analytical method to find these modal shapes in this particular case (not another one) or an analytical solution if it exists!! (I did some analytical development using the Homotopy Analysis Method but I obtained a not very "elegant"/"beautiful" expression of the solution , especially if I want to compute it, I think it is a little bit complicated as expression!! )
I am not also interested also in simple sinusoids solutions (the classic procedure to approach modal shapes).
Thank you.
(Attached the equation with the variables in a pdf file : equation to solve.pdf)
Dear Mr Mansour,
Differential equation is a very general notion: linear, non-linear, order of the differential equation, and so on. In order to recommend a solution method, you need to give us the concrete equation and then consider how it can be solved.
Best regards
K. Yankov
Dear Mr. Yankov,
The equation is defined in the file equation tosolve.pdf attached in my question.
Regards.
A.Mansour
Dear Mr. Liu,
Thank you for your answer, but I think that the remark you mentioned it is only valid when the second member "F" is constant. In the other case, when "F" is a space-dependent function (which is the case I am considering), the method you proposed does not hold any more as I think (the derivatives of the new function y(s) will contain derivatives of F also and then we will have a new inhomogeneous differential equation with variable coefficients, so we will return to the first case !! )
Thanks.
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