dy/dx=a^2(1+y^2)-f(x)y is a first order nonlinear ODE. Its numerical solution can be made using an initial condition. If you send me a boundary condition, I can send you numerical solution of it

Thank you Dr, it is a kind of Riccati equation and I am looking for an analytical method to solve it.

It is very unlikely that there is an analytic method which works for all f. I only know the analytic solution for a=1 and f=0.

I solved this equation for several choice of f(x) with a^2=1. For example, f(x)=constant or f(x)=x... I think we need a proper change of variable. Although, y=Tan u seem to be a good choice but it did not work. You try to see where we get a small difficulty. Hence I am sure there must be a proper change of variable.

Saravi, I have just found a starting point for your problem. I think that the exponential function is the best candidate. You find attached a short description.

One should now look for eventual solutions of this equation which is simpler than the first one!

Dear Dr, I did try before and I got the same difficulty as choosing y=Tan u . How ever thanks

Saravi. The homogeneous solution can be found analytically. It is also possible to find a particular one. I think that the solution of the general problem hides behind a combination of these two solutions. What do you think? I have no more ideas about this nonlinear differential equation!

@Abdallah

Athough you realize that the ODE is non-linear, your strategy is based on expectations that are justified only for linear equations.

@M. Saravi

By separation of variables one easily solves the special case f=const ( for arbitrary a). This is a valid special case of your general problem and thus gives you a lower limit for the potential complications of the general solution. Further, it provides some intuition by comparing it with motion of systems acted upon by time-dependent forces. If for such systems the motion for stationary forces is known one may find ways to build the motion under time variant forces.

@all

Isn't it again and again surprising that finding a solution by numerical methods is so easy (here the non-linearity does not hurt at all) ?

The simplest way to solve this equation numerically is to apply the 1st order Euler scheme. This approach is indeed 1st order consistent. In order to obtain convergence to the exact solution we have to check also its stability to small perturbations of the initial condition. It seems that it is not possible to check this stability due to the non-linearity and the numerical sequence can't be expressed in terms of the initial value.

Abdallah,

you see a limitation of the numerical method which does not exist. Simply vary the initial conditions and see what happens! By the way: there are explicit methods which are nearly as simple as Euler and that are of second order and thus much more practicable. A second BTW: Proving that a ODE satisfies the Lipschitz condition is easily possible also for many non-linear ODEs, then local well-behavior of solutions is no problem also from a theoretical point of view.

Yes, that is true but I am insisting for analytical method. Because I am working on a larger problem that a part of it is this equation. In fact, I have not any initial value and can not force any initial value to this or main problem.

I send my answer as a file attachment only in the case if the function f(x) is absolutely analytically function. Please, answer me. Thank You, prof. Ulrich Mutze. Your sincerely, Anna Tomova.

I send my answer another way with any corrections about the values of the coefficients as a file attachment only in the case if the function f(x) is absolutely analytically function. Please, answer me. Thank You, prof. Ulrich Mutze. Your sincerely, Anna Tomova.

@Anna

Great that you did not forget what a century ago belonged to the basics for each mathematician, physicist, and engineer. Riccati's equation (and its generalization by d'Alembert) was among the first mathematical topics that fascinated me as a pupil. Unfortunately the stuff did not return into my mind when thinking about Masoud's question. Your achievement is in particular to have shown that power-series ansatz leads to a managable iteration scheme for a particular solution. If this is obtained, the rest is good-old stuff, see e.g. E.L.Ince: ordinary differential equations Dover 1956 (Original 1926).

Thanks for all suggestions sending me.  Still I was not satisfied with these methods. I thought for start it may be better to find some f(x) which leads to an analytical solution. For example, write y'=1-fy+y^2 in the form of y^2-fy+1-y'=0 and solve this second degree equation for y. One can shows f=2tanh(x) then y=tanh(x). If this equation has double roots then we get y'=1 -f^2/4, which can be true for some f(x) not all. I was busy to finish my second book on solution of nonlinear ODEs. Fortunately, we completed it and been published on June 2016. Now, I got more time to spend on solution of this equation. I am not sure I can complete it but during this research I could obtain few solution for solving nonlinear ODEs that I put in my book. Once more thank you all.