Finding the Second Solution for a System with a Repeated Eigenvalue

In summary, the conversation discusses solving a given matrix equation with a repeated eigenvalue of 2 and only one eigenvector. The solution is found using a specific procedure, resulting in the equation X(t) = ce^(2t)(1,1) + de^(2t)(t,1). The process involves finding a second solution using the eigenvector and a constant vector.
  • #1
Buri
273
0

Homework Statement



I'm told to solve the following:

X' = (1 1; -1 3)X

Where (1 1) is the first row and (-1 3) is the second row.

The Attempt at a Solution



Okay so I calculated teh eigenvalues and I got repeated eigenvalue of 2. I calculated the eigenvector and got (1,1). So a solution would be X(t) = ce^(2t)(1,1).

My text considers the matrix (a 1; 0 a) before the problems and goes on to find a eigenvalue with eigenvector, but then adds another solution to it so getting:

X(t) = ce^(at)(1,0) + de^(at)(t,1)

I understand how they got it in that one, but I'm not exactly sure how to go about finding one for mine. Any help?
 
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  • #2
There's a procedure for this. If a system [itex]\vec{X}^{\prime}=A\vec{X}[/itex] has a repeated eigenvalue [itex]\lambda[/itex] and there is only one eigenvector [itex]\vec{K}[/itex] associated with it (as you have here), then the second solution [itex]\vec{X}_2[/itex] that you seek is found as follows.

[tex]\vec{X}_2=\vec{K}te^{\lambda t}+\vec{P}e^{\lambda t}[/tex],

where [itex]\vec{P}[/itex] satisfies:

[tex]\left(A-\lambda I\right)\vec{P}=\vec{K}[/tex].
 

FAQ: Finding the Second Solution for a System with a Repeated Eigenvalue

What is a differential equation?

A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is used to model and describe various physical phenomena in fields such as physics, engineering, and economics.

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Differential equations are important because they allow us to understand and predict how a system will change over time. They are often used in scientific and engineering fields to model complex systems that cannot be represented by simple algebraic equations.

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There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve only one independent variable, PDEs involve multiple independent variables, and SDEs involve random variables.

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