Uncoupling Equations: A Matrix Approach

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Homework Help Overview

The discussion revolves around uncoupling a system of differential equations involving a linear equation and a nonlinear equation. The subject area includes differential equations and matrix methods for solving such systems.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the possibility of using matrix methods to uncouple the equations, while questioning the applicability due to the nonlinear nature of one of the equations. There is a suggestion to solve for one variable in terms of another and substitute it into the second equation. Additionally, participants discuss the form of the solution for t as a function of s and its implications for u.

Discussion Status

The discussion is active, with participants providing insights and suggestions on how to approach the problem. Some guidance has been offered regarding the substitution method, and there is an ongoing exploration of the implications of the proposed solutions.

Contextual Notes

There is a mention of the nonlinear nature of one of the equations, which may affect the methods used for uncoupling. Participants are also encouraged to verify their own calculations and assumptions regarding the relationships between the variables.

Winzer
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Homework Statement


I am having trouble remembering how to uncouple these.



Homework Equations


[tex]\frac{dt}{ds}=1[/tex]
[tex]\frac{du}{ds}=2tu[/tex]



The Attempt at a Solution


I remember putting it into a matrix.
[tex]x'=\lambda x[/tex]
 
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The equation with du/ds is nonlinear, so this system might not be amenable to uncoupling by matrix methods, which involves finding eigenvalues and eigenvectors, and using them to diagonalize a matrix.

Alternatively, I think it works to solve for t as a function of s in the first equation, and substitute for t in the second equation, and solve it for u.

To get you started, if dt/ds = 1, what is t as a function of s? Hint: there is not just one solution.
 
t(s)=s+C

Plugging that into the latter u = A Exp(s^2+Cs). Is that right?
 
Winzer said:
t(s)=s+C

Plugging that into the latter u = A Exp(s^2+Cs). Is that right?

Check it your self! Certainly dt/ds= 1. What is du/ds? Is it equal to 2tu?
 

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