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

In summary, the group discusses a system of first-order ordinary differential equations and its exact solutions. They mention the use of the quotient of two equations and suggest investigating the behavior of solutions through phase portraits. They also mention the impossibility of constructing an exact solution for the system in full generality.
  • #1
googler
7
0
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
 
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  • #2
I assume you mean a4 in the 2nd equation, can't you just look at the quotient of the two?
 
  • #3
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.
 
  • #4
Well what does y'/x' look like? Note that you are dropping a solution.
 
  • #5
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.
 
  • #6
No... if y' = dy/dt and x' = dx/dt then y'/x' = ?
 
  • #7
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?)
 
  • #8
... why doesn't it work in that case? x = sin(sqrt(y)) or x = sin(-sqrt(y))
 
  • #9
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.
 
  • #10
Well it looks like it can be made into an exact ODE so I'd go that route probably
 
  • #11
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.
 
  • #12
Hi Kenneth,

googler said:
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.
 
  • #13
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 :)
 
  • #14
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!

Unco said:
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.
 

1. What is a system of ODE?

A system of ODE (ordinary differential equations) is a set of equations that describe the change of a variable with respect to another variable, where the change is dependent on the value of the variable itself. These equations are typically used to model dynamic systems in various fields such as physics, engineering, and biology.

2. How do you solve a system of ODE?

The most commonly used method to solve a system of ODE is through numerical methods, such as Euler's method or Runge-Kutta methods. These methods involve breaking down the system into smaller time intervals and using iterative calculations to approximate the values of the variables at each time step. Alternatively, some systems of ODE may have analytic solutions that can be derived using mathematical techniques.

3. What are the applications of solving systems of ODE?

Solving systems of ODE has many real-world applications, including predicting the behavior of physical systems, analyzing chemical reactions, modeling population dynamics, and understanding the spread of diseases. It is also used in fields such as economics, finance, and neuroscience to study complex systems.

4. What are the challenges in solving systems of ODE?

One of the main challenges in solving systems of ODE is finding an accurate and efficient numerical method that can handle the complexity of the system. Another challenge is ensuring that the initial conditions and parameters used in the equations are realistic and reflect the real-world system being modeled. Additionally, some systems may have chaotic behavior, making it difficult to accurately predict their future behavior.

5. How can I check the accuracy of my solution to a system of ODE?

The accuracy of a solution to a system of ODE can be checked by comparing it to known analytical solutions, if available, or by using convergence tests to determine how closely the numerical solution is approaching the true solution. It is also important to ensure that the numerical method used is appropriate for the specific system being solved and that the initial conditions and parameters are accurate.

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