I suppose that by solving a differential equation, you mean to find an exact solution. In many of my research areas, we need to find exact solutions, not just an approximation. Unfortunately, as far as I know there are no other good general methods to solve differential equations of higher order. In fact, even the classical undetermined coefficient method, it only provides approximations in most cases, not exact solutions.

Hello,  the original question is not precise.

Do you mean linear higher order differential equations?

In any case provide us with an example, the coefficients of which are polynomials.

Regards,

Gabriel

@Thomas  I am asking about linear higher order differential equations.

As Bang - Yen Chen has written, there are no general methods for solving exactly such linear differential equations involving arbitrary polynomial coefficients. A general method of approximating the solution is to attach an appropriate first order linear differential system and to apply a fixed point type argument to the latter system.

If your research is apllied and you need only numerical solutions, a good way of brute force would be any stochastic optimization method. Good luck!

Since you are in management line , I do not know how to suggest you.

Some suggestions

(i) try to convert it in to two second order equations. then solve it.

(ii) Read the book in Mathematical physics by B.D.Gupta ,hope you will understand

your problem.

(iii) write your equation directly in your question,so that others will help you .

(iv) Take help of some people having knowledge in MATLAB .

Good luck

http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-UseSeries-SolveDEs_Stu.pdf

First look  this article. And tell mi is that you where asking for.

@Borut Jurcic Zlobec

Yes. I was finding a method to solve LDE involving polynomials as coefficients. But I think solve DE using Power Series isn't a better way as we have to approximate the power series in the solution because " we are not usually able to express power series solutions of differential equations in terms of known functions".

If it is not possible to express the solution in terms of known functions you could'n do it by any other method. In this case you have new functions. But in real life the  differential equation  generally will be solved numerically..

https://people.maths.ox.ac.uk/suli/nsodes.pdf

Try with semi analytical method known as Homotopy Analysis method(HAM).

There is beautiful Sturm Louville theory about special kind of second order ordinary differential equations where mathematical physics begins. 

http://www.iitg.ernet.in/physics/fac/charu/courses/ph402/SturmLiouville.pdf

Eventhough the coefficients are polynomials we can apply Legendre and Chebyshev wavelet methods to solve such differential equations.