A system of DEs with variable coefficients.

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Discussion Overview

The discussion revolves around a system of differential equations (DEs) with variable coefficients. Participants explore the possibility of solving these equations analytically, while also considering numerical methods and the implications of specific terms within the equations. The context includes initial conditions provided by the original poster.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • The original poster expresses difficulty in solving the system of DEs analytically and questions whether it is possible.
  • One participant suggests that the nonlinear nature of the equations indicates a shift towards numerical solutions.
  • Another participant notes that the derivative ##z'(t)## starts at -3 and raises concerns about the behavior of the solutions, specifically mentioning a blow-up after 1.6 seconds.
  • The original poster mentions having solved the equations numerically in Mathematica and seeks an approximate analytical solution, while also considering the potential neglect of the ##x^2(t)## term in the equation for ##y'(t)##.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the feasibility of an analytical solution, with some advocating for numerical methods and others exploring the analytical approach. The discussion remains unresolved regarding the best method to tackle the system of DEs.

Contextual Notes

The discussion highlights the complexity of the nonlinear system and the potential for divergent behavior in the solutions, particularly concerning the initial conditions and the terms involved in the equations.

inertiagrav
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Hi. I have been trying for sometime to solve the following system of DEs analytically(Is it possible?) but no luck so far.
$$x''(t)=-z(t)x'(t)-x(t)+y(t),$$ $$y'(t)=-z(t)y(t)+x^2(t)$$ $$z'(t)=-2z^2(t)-x(t)$$.

With the initial conditions ##x(0)=1## , ##x'(0)=0## ,##y(0)=0## and ##z(0)=1##.

Thanks a lot in advance.
 
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nonlinear... start looking at a numerical solution
 
##z'(t)## starts at -3 and things blow up after 1.6 seconds. What do these equations represent ? How long is it supposed to run ?
 
Hello, thanks for your responses. I solved it numerically in Mathematica before posting it here, even for different sets of initial conditions. I am trying to get an approximate analytical solution and yea, i will ask for a context regarding the equations. Our Professor did say that if needed we can neglect the ##x^2(t) ## term in ##y'(t)## but i still don't see how i can solve it.
 

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