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January 16th, 2012, 03:06 PM   #1
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Solve complicated differential equation

I have to solve the following differential equation:



where you assume solutions of the form:



, , , are constants and is the imaginary unit.

Could someone please tell me how to obtain explicit expressions for , , , ?

I got as far as rewriting the assumed form for in sines and cosines with Euler's rule. Then I differentiated that form and set it equal to . I also rewrote in terms of sines and cosines. But I do not see how to obtain explicit expressions.

I need to hand this in by Friday...
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January 16th, 2012, 09:51 PM   #2
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Re: Solve complicated differential equation

It will be much easier with :
z = r exp(i a)
where r(t) and a(t) are real
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January 17th, 2012, 12:12 AM   #3
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Re: Solve complicated differential equation

That form might make things easier, however you are also simplifying the problem. Using that form you are assuming harmonic solutions without exponential growth or decay. This problem is taken from a textbook (Nonlinear differential equations and dynamical systems, Ferdinand Verhulst). The reason why I need to have the explicit expressions is to know whether , or . With that information I can answer the question, whether the periodic solutions are bounded or not. The solutions in the back of that same book hints at taking that form for solving the problem. It also states that the solutions are not bounded. Unfortunately no work-out was added.
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January 17th, 2012, 10:11 AM   #4
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Re: Solve complicated differential equation

The analytical solution is presented in the joint document.
The solution is expressed on a parametric form.
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January 17th, 2012, 12:25 PM   #5
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Re: Solve complicated differential equation

Thank you JJacquelin. You have put a lot of work into it. However as I have said before, I need the solution in the form that I proposed:



By doing it your way, you are simplifying the problem by assuming that the final solutions will not contain any exponential growth or decay.
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January 17th, 2012, 12:28 PM   #6
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Re: Solve complicated differential equation

As a reminder. I do not need a solution for . I only need expressions for the constants , , and .
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January 17th, 2012, 01:33 PM   #7
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Re: Solve complicated differential equation

Quote:
Originally Posted by Caluca
As a reminder. I do not need a solution for . I only need expressions for the constants , , and .
Since it is necessary to express the solution on a parametric form in order to involve only usual functions, I think that what you expect is not possible. I mean that p, q, mu and nu probably cannot be expressed in terms of a combination of a finite number of usual functions.
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January 18th, 2012, 06:57 AM   #8
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Re: Solve complicated differential equation

Here is my work-out. What I am aiming at is to get one term at each side. Then you can compare the left and right exponent and the left and right coefficient. This gives 2 equations. Then by splitting them up into real and imaginary parts and comparing the left ones with the right ones, you'll get four equations and four unknowns. However I am unable to get there. Help someone?
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January 18th, 2012, 11:55 AM   #9
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Re: Solve complicated differential equation

You are looking for something impossible.
Of course, the differential equation has analytic solutions, but NONE on the form that you expect. A very simple proof is given in the attachment.
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January 19th, 2012, 04:29 PM   #10
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Re: Solve complicated differential equation

Sorry Jacuellin for being stubborn, but I have to disagree with you, because I have found the solution. You were on the right track with your proof. However you took a wrong conclusion. You are right by saying that it is impossible to have the left hand side being time dependent, while the right hand side is a constant. So the way to proceed is to make sure that the left hand side is NOT time dependent. But first let me correct in error in your last equation; it should be:



You can ensure that the left hand side is not time dependent by setting:



This results in:

Thus now you get:







Now you obtain a system of two equations by ensuring that the real part of the left hand side is equal to the real part of the right hand side and likewise for the imaginary parts:





Substitution of the first one into the second one and simplifying gives:



Simplifying and rearranging leads to:





The value of was already calculated to be equal to

Thus:



The solution is now a superposition of twice the proposed form, once with either values of . Because the solution contains at least one real component exponent which is positive, the solutions are not bounded.

Q.E.D.

The reason why I could not have solved this earlier is because I did not know that was the correct complex conjugate. That is why I first expanded into sines and cosines in my attachment. Thank you for your interest Jacuellin!
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