What is the general solution of x^{(k)}+(\cos t + \sin\sqrt{2}t ) x = 0 ordinary differential equation for k>=2?
Dear Rushan Ziatdinov ,
((d^{k}x)/(dt^{k}))+(cos t+sin√2t)x = 0.
Solution: Let k = 3
x₁ = x(t)
x₂ = dx(t)/dt
x₃ = d²x(t)/dt²
This will transform the ODE into the homogeneous non- autonomous system:
dx₁/dt = x₂
dx₂ /dt = x₃
dx₃/dt = - (cos t+sin√2t)x₁
Now, with given initial conditions, we write the system in matrix form
and find a general solution using the usual method.
(See the attached article).
Similarly, you can choose any positive integer value for k > 2.
Best regards
Dear Paul Marcoux
,To solve such type of higher order ODE's, we transform the equation into
a system of non-autonomous DE's.
And then we solve the new system using the matrix of coefficients A(t).
Do you think my proposed answer will not work?
Notice that the matrix coefficients include cos\sin function and constants
that turns A(t) into a periodic matrix for some period to be determined. Check the attached file in my previous answer.
Best regards
Dear Paul Marcoux
,Exact explicit solutions are not the goal of such systems.
Using the system, we have the solution x(t) = 0 is an equilibrium state of the dynamical system represented by the non -autonomous system. We need to study the stability of the equilibrium state.
In the absence of the explicit solutions ( Direct method: based on the real parts of the eigenvalues) so, we need to use the indirect method of Lyapunov. The periodicity of A(t) helps to use the last part of the mentioned article that focus on this point. ( See Thm 3.36).
Best regards
Dear all,
thank for your messages. I have found this problem in the following manuscript (page 167)
Russian Mathematical Surveys, 1989, 44:4, 157–171
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=1854&option_lang=eng
This equation reminds me so-called Hill differential equation.
https://en.wikipedia.org/wiki/Hill_differential_equation
You can see Piecewise analytic method (PAM)-(Published January 2012 by T. Abassy) is a new technique used for solving nonlinear differential equation. The advantages of PAM are( these points are proofed):
1. It gives a general analytic formula that can be used in differentiation and integration.
2. It can solve highly non-linear differential equation.
3. The accuracy and error can be controlled according to our needs very easily.
4. It can solve problems which other famous techniques can’t solve.
5. In some cases, it gives the exact solution.
You can follow the updates and discussion and comments on the URL:
https://www.researchgate.net/post/What_do_you_know_about_Piecewise_Analytic_Method_PAM-Published_January_2012_by_T_Abassy
will contain all advantages and disadvantages of PAM.
Dear Tamer Ahmed Abassy ,
You are describing your proposed method to solve such differential equations.
Approximations are well-known to find (approximate) solutions in some neighborhoods. It is appreciated if you can show how to solve this k^th - Order ODE equation using your approach.
Best regards
Dear Issam Kaddoura
This is a quick solution using
x(0)=.01
dx(0)/dt=0
k=2
see the attached file and give me your opinion.
Dear Tamer Ahmed Abassy ,
Thank you for your response to my request.
I think you have implemented (some numerical algorithm) to reach the posted graph. Notice that the ODE is linear of order k.
(1) You mentioned the step size h = 0.01 and the order k = 10, but you said nothing about the necessary 10 initial conditions.
Knowing that different choices will produce different answers!
(2) The equilibrium state of the corresponding dynamical system is at X(t) = 0.
Unfortunately, this graph can say nothing about the stability of the equilibrium state. Did you start from some initial point close to X(t) = 0?
Another question:
Do you think that your (Algorithm) can discover limit cycles if any?
Neighborhoods are of great importance to give the correct answer.
Best regards
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