Solving System of First-Order ODEs: Exact Solution for x(t)

[googler]

Dear all,

I have been trying to solve the following system of first-order ordinary differential equations for a week:
x' = y * (a1*x + a2*y + c1),
y' = y * (a3*x + a4*y + c2),
where x and y are functions of t, and ai and ci are constants. This system seems not very complex, but I have not found the its exact solutions.

Can anyone tell me the exact solution for x(t)? Many thanks!Kenneth

 

[NoMoreExams]

I assume you mean a4 in the 2nd equation, can't you just look at the quotient of the two?

 

[googler]

Hi 'NoMoreExams', thank you very much. The typo has been corrected.

Could you tell me how you solve this problem by making use of the quotient? Many thanks for your patience and time.

 

[NoMoreExams]

Well what does y'/x' look like? Note that you are dropping a solution.

 

[googler]

I tried this way... y'/x' = (a3*x + a4*y + c2)/(a1*x + a2*y + c1), and we could find some relationship between x*y', y^2, x'*y, x^2, x', and y'. But what follows? Sorry, I am confused.

 

[NoMoreExams]

No... if y' = dy/dt and x' = dx/dt then y'/x' = ?

 

[googler]

That's cool. But does that mean y MUST BE A FUNCTION of x? Originally this does not necessarily hold. (For example, can we use this method if x = sin(t) and y = t^2?)

 

[NoMoreExams]

... why doesn't it work in that case? x = sin(sqrt(y)) or x = sin(-sqrt(y))

 

[googler]

You are very smart :)
I am not sure if I am right now, because it becomes more and more complicated as I continue this procedure... Could you tell me the result, if it is simple to you? Thanks once again.

 

[NoMoreExams]

Well it looks like it can be made into an exact ODE so I'd go that route probably

 

[NoMoreExams]

You should also try to go about this in a different way, for your own education. Find the fixed points, figure out their stability and try to draw the phase portrait yourself.

 

[Unco]

Hi Kenneth,

Dear all,

I have been trying to solve the following system of first-order ordinary differential equations for a week:
x' = y * (a1*x + a2*y + c1),
y' = y * (a3*x + a4*y + c2),
where x and y are functions of t, and ai and ci are constants. This system seems not very complex, but I have not found the its exact solutions.

Can anyone tell me the exact solution for x(t)?

For the system of ODEs as you have them in full generality, one cannot hope to write the nontrivial solutions in terms of elementary functions (a computer package-generated solution involving the Lambert W function et al instructs me as much). Investigating the behaviour of solutions by way of phase portraits, as NoMoreExams suggested, would seem more appropriate. Indeed, performing this kind of analysis for various values of the constants hints at the impossibility of constructing an exact solution.

 

[googler]

Hi Unco and 'NoMoreExams',

Thank you very much for your help. Actually I am just interested in the case where c2 = -c3. I have obtained the numerical solution using MATLAB but it is not enough since I need to find the property of all possible x or \int x dt. Now I am trying the way of phase portraits :)

 

[googler]

Hi Unco,

Could you please tell me what package you used to generate the solution involving the Lambert W function (and the solution you found)? It is OK that the solution could not be written in terms of elementary function. I just need to analyze the behavior of the solution.

Many thanks!

Hi Kenneth,


For the system of ODEs as you have them in full generality, one cannot hope to write the nontrivial solutions in terms of elementary functions (a computer package-generated solution involving the Lambert W function et al instructs me as much). Investigating the behaviour of solutions by way of phase portraits, as NoMoreExams suggested, would seem more appropriate. Indeed, performing this kind of analysis for various values of the constants hints at the impossibility of constructing an exact solution.