Graduate Solution to a differential equation with variable coefficients

[mertcan]

Hi, I really struggled to dig valuable things out of internet and books related to high order homogeneous differential equation with variable coefficients but I have nothing. All methods I see involves given solution and try to find others(like reduction of order method), even for second order homogeneous differential equation with variable coefficients case methods involves a known solution. My question is: WITHOUT ANY GIVEN SOLUTION how can we derive the solutions or just a solution of high order homogeneous differential equation with variable coefficients?

 

[jedishrfu]

My guess would be numerically for the hardest of these as textbooks tend to focus on ones with exact solutions or ones that fall to known methods of solution. It is perhaps why MATLAB and related tools are so popular today in engineering fields.

 

[mertcan]

My guess would be numerically for the hardest of these as textbooks tend to focus on ones with exact solutions or ones that fall to known methods of solution. It is perhaps why MATLAB and related tools are so popular today in engineering fields.

Thanks for return but as far as I know numerical methods are aimed at finding some value at some t using finite difference schemes..., not finding a function with respect to t. My question is: WITHOUT ANY GIVEN solution (as a function of t) how we can find the solution(s) of variable coefficient high order homogenous differential equation(as a function of t)?

 

[jedishrfu]

Curiously there's Eureqa when you have data but no idea what the function is. However, i imagine that's not what youre looking for.

https://en.m.wikipedia.org/wiki/Eureqa#Origin_and_Development

 

[bigfooted]

There are no algorithms for finding even the general solution of a second order homogeneous ODE with variable coefficients. There is however an algorithm to find the solution if the solution is Liouvillian, which is based on differential Galois theory and Picard-Vessiot theory, this is the Kovacic algorithm. This is the algorithm that e.g. wolfram, mathematica, maple etc. uses to symbolically solve homogeneous ODEs. Kovacic method has been extended to higher order ODEs as well.

 

[mertcan]

There are no algorithms for finding even the general solution of a second order homogeneous ODE with variable coefficients. There is however an algorithm to find the solution if the solution is Liouvillian, which is based on differential Galois theory and Picard-Vessiot theory, this is the Kovacic algorithm. This is the algorithm that e.g. wolfram, mathematica, maple etc. uses to symbolically solve homogeneous ODEs. Kovacic method has been extended to higher order ODEs as well.

Thanks for return@bigfooted by the way do you have any idea about WITHOUT ANY GIVEN SOLUTION how can we derive the solutions or just a solution of high order homogeneous differential equation with variable coefficients?

 

[bigfooted]

Well, that is what the Kovacic algorithm does for instance. For second order linear ODE's, you can prove that all Liouvillian solutions can be found by finding rational solutions to a Riccati ODE. And finding these solutions can be done in a systematic way by solving systems of polynomial equations. But these algorithms are usually too lengthy to do by hand, that is why we only learn about some simple limit cases in any course on differential equations (like constant coefficient ODEs).