A system of 1st order nonlinear differential equations

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Discussion Overview

The discussion revolves around a system of first-order nonlinear differential equations, focusing on the interpretation of the equations, potential methods for solving them, and the mathematical validity of certain expressions within the system.

Discussion Character

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

Main Points Raised

  • Participants seek suggestions for solving a system of first-order nonlinear differential equations.
  • Some participants question the meaning of "u^2" in the equations, suggesting it may refer to a dot product or a cross product, but express confusion over its mathematical implications.
  • A participant notes that if "u^2" is a dot product, it cannot be multiplied by a two-by-two matrix, and if it is a cross product, it cannot be added to a matrix like C(t).
  • Another participant clarifies that "u^2" is intended as a cross product, providing a specific representation of it.
  • Suggestions for numerical solutions include the Runge-Kutta method, with a request for potential analytical solutions if they exist.
  • There is a request for clarification on whether an analytical solution is possible for the given system.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of "u^2" and its mathematical validity, indicating a lack of consensus on this aspect of the equations. The discussion on solution methods remains open-ended, with no agreement on the feasibility of an analytical solution.

Contextual Notes

Participants have not settled on the definitions of certain terms or the mathematical operations involved, leading to ambiguity in the discussion. The potential for both numerical and analytical solutions is mentioned, but no definitive methods are agreed upon.

quacam09
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Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

<br /> <br /> \[<br /> \begin{array}{l}<br /> u&#039;(t) = Au^2 (t) + B(t)u + C(t) \\ <br /> u(t) = \left[ {\begin{array}{*{20}c}<br /> {x_1 (t)} \\<br /> {x_2 (t)} \\<br /> \end{array}} \right] \\ <br /> A = \left[ {\begin{array}{*{20}c}<br /> {a_{11} } &amp; {a_{12} } \\<br /> {a_{21} } &amp; {a_{22} } \\<br /> \end{array}} \right] \\ <br /> B(t) = \left[ {\begin{array}{*{20}c}<br /> {f_{11} (t)} &amp; {f_{12} (t)} \\<br /> {f_{21} (t)} &amp; {f_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> C(t) = \left[ {\begin{array}{*{20}c}<br /> {g_{11} (t)} &amp; {g_{12} (t)} \\<br /> {g_{21} (t)} &amp; {g_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> \end{array}<br /> \]<br /> <br />
 
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quacam09 said:
Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

<br /> <br /> \[<br /> \begin{array}{l}<br /> u&#039;(t) = Au^2 (t) + B(t)u + C(t) \\ <br /> u(t) = \left[ {\begin{array}{*{20}c}<br /> {x_1 (t)} \\<br /> {x_2 (t)} \\<br /> \end{array}} \right] \\ <br /> A = \left[ {\begin{array}{*{20}c}<br /> {a_{11} } &amp; {a_{12} } \\<br /> {a_{21} } &amp; {a_{22} } \\<br /> \end{array}} \right] \\ <br /> B(t) = \left[ {\begin{array}{*{20}c}<br /> {f_{11} (t)} &amp; {f_{12} (t)} \\<br /> {f_{21} (t)} &amp; {f_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> C(t) = \left[ {\begin{array}{*{20}c}<br /> {g_{11} (t)} &amp; {g_{12} (t)} \\<br /> {g_{21} (t)} &amp; {g_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> \end{array}<br /> \]<br /> <br />

What does u2(t) mean? Is it u(t) \cdot u(t)?

Also, shouldn't the differential equation be
u&#039;(t) = Au^2 (t) + B(t)u(t) + C(t)
?
 
quacam09 said:
Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

<br /> <br /> \[<br /> \begin{array}{l}<br /> u&#039;(t) = Au^2 (t) + B(t)u + C(t) \\
<br /> As Mark44 notes, the &quot;u^2&quot; doesn&#039;t make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn&#039;t make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> u(t) = \left[ {\begin{array}{*{20}c}<br /> {x_1 (t)} \\<br /> {x_2 (t)} \\<br /> \end{array}} \right] \\ <br /> A = \left[ {\begin{array}{*{20}c}<br /> {a_{11} } &amp; {a_{12} } \\<br /> {a_{21} } &amp; {a_{22} } \\<br /> \end{array}} \right] \\ <br /> B(t) = \left[ {\begin{array}{*{20}c}<br /> {f_{11} (t)} &amp; {f_{12} (t)} \\<br /> {f_{21} (t)} &amp; {f_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> C(t) = \left[ {\begin{array}{*{20}c}<br /> {g_{11} (t)} &amp; {g_{12} (t)} \\<br /> {g_{21} (t)} &amp; {g_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> \end{array}<br /> \] </div> </div> </blockquote>
 
For some reason, HallsOfIvy's reply didn't render correctly. Here it is.
HallsofIvy said:
As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix.
 
Mark44 said:
For some reason, HallsOfIvy's reply didn't render correctly. Here it is
As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix..

Thanks for your help and sorry for unclear things.

u^2 is a cross product. It means
<br /> <br /> <br /> \[<br /> u^2 (t) = \left[ {\begin{array}{*{20}c}<br /> {x_1^2 (t)} \\<br /> {x_2^2 (t)} \\<br /> \end{array}} \right] \\ <br /> <br /> \]<br />

And C(t)
<br /> C(t) = \left[ {\begin{array}{*{20}c}<br /> {g_{1} (t)} \\<br /> {g_{2} (t)} \\<br /> \end{array}} \right] \\ <br /> <br />
 
EnumaElish said:
For a numeric solution the Runge-Kutta method seems an especially good method to use: http://www.springerlink.com/content/w080u7262137j867/

Thanks EnumaElish. Is an analytical solution impossible? If there is a method to obtain an analytical solution, can you suggest me?
 

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