Uncoupling Equations: A Matrix Approach

In summary, the conversation discusses strategies for uncoupling a system of equations involving \frac{dt}{ds}=1 and \frac{du}{ds}=2tu. One method involves using matrix methods to find eigenvalues and eigenvectors, while another involves solving for t as a function of s and substituting it into the equation for u. The conversation also provides a hint for solving for t as a function of s.
  • #1
Winzer
598
0

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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  • #2
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.
 
  • #3
t(s)=s+C

Plugging that into the latter u = A Exp(s^2+Cs). Is that right?
 
  • #4
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?
 

Related to Uncoupling Equations: A Matrix Approach

1. What is the purpose of uncoupling equations?

Uncoupling equations are used to solve systems of equations that are coupled or interconnected. By using a matrix approach, we can separate these equations and solve them individually, making the overall problem easier to solve.

2. How do you represent uncoupling equations in matrix form?

The uncoupling equations are represented by creating a matrix with the coefficients of the variables in the equations. The right-hand side of the equations are also included in the matrix as a column vector. By using matrix operations, we can manipulate this matrix to solve for the variables.

3. Can uncoupling equations be used for non-linear systems?

Yes, uncoupling equations can be used for both linear and non-linear systems. However, the process of solving non-linear systems may be more complex and may require additional techniques such as iteration or approximation.

4. What are the advantages of using a matrix approach for uncoupling equations?

Using a matrix approach allows us to easily manipulate and solve systems of equations, regardless of the number of variables. It also allows us to solve larger systems in a more efficient and organized manner, compared to solving each equation individually.

5. Are there any limitations to using uncoupling equations?

Uncoupling equations may not always be the most efficient or accurate method for solving systems of equations. In some cases, other techniques such as substitution or elimination may be more suitable. Additionally, uncoupling equations may become more complex and time-consuming for systems with a large number of variables.

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