Solve Eigenvalue Method for dx/dt=[12 -6; 6 -3] w/x(0)=[12; 9]

In summary, the problem is to solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]. To solve this, the first step is to find the Eigenvalues, which can be obtained by setting the determinant of the coefficient matrix equal to 0. In this case, the Eigenvalues are λ=3 and λ=-3. However, the attempt at a solution provided is incorrect and the correct method for finding Eigenvalues should be shown.
  • #1
jrsweet
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Homework Statement


solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]


Homework Equations





The Attempt at a Solution


I know I need to find the Eigenvalues but then I get a little confused from there.

(λ-3)(λ+3)=0
λ=3, -3
 
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  • #2
jrsweet said:

Homework Statement


solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]
A assume you mean "dx/dt= [12 -6; 6 -3]x" or, in Latex,
[tex]\frac{dX}{dt}= \frac{d\begin{bmatrix}x \\ y\end{bmatrix}}{dt}= \begin{bmatrix}12 & -6 \\ 6 & -3\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex]


Homework Equations





The Attempt at a Solution


I know I need to find the Eigenvalues but then I get a little confused from there.

(λ-3)(λ+3)=0
λ=3, -3
That's wrong. Show us how you got the eigenvalue equation.
 

What is the Eigenvalue Method?

The Eigenvalue Method is a mathematical technique used to solve systems of linear ordinary differential equations. It involves finding the eigenvalues and eigenvectors of a matrix, which can then be used to find the general solution to the system.

How is the Eigenvalue Method used to solve dx/dt=[12 -6; 6 -3] w/x(0)=[12; 9]?

In order to solve this system using the Eigenvalue Method, we first need to find the eigenvalues and eigenvectors of the matrix [12 -6; 6 -3]. This can be done by solving the characteristic equation det(A-λI)=0. Once we have the eigenvalues and eigenvectors, we can use them to construct the general solution to the system.

What are the eigenvalues and eigenvectors of [12 -6; 6 -3]?

The eigenvalues of [12 -6; 6 -3] can be found by solving the characteristic equation det(A-λI)=0. In this case, the eigenvalues are λ=0 and λ=15. The corresponding eigenvectors are [1; 2] and [1; -1] respectively.

How do I use the eigenvalues and eigenvectors to find the general solution?

Once we have the eigenvalues and eigenvectors, we can use them to construct the general solution to the system dx/dt=[12 -6; 6 -3] w/x(0)=[12; 9]. This can be done by writing the solution in the form x(t)=c1e^(λ1t)v1+c2e^(λ2t)v2, where c1 and c2 are constants and v1 and v2 are the eigenvectors corresponding to the eigenvalues λ1 and λ2 respectively.

What are the initial conditions for this system and how do I use them in the solution?

The initial conditions for this system are x(0)=[12; 9]. This means that at t=0, the values of x1 (the first element in the vector x) and x2 (the second element in the vector x) are 12 and 9 respectively. To use these initial conditions in the solution, we can substitute t=0 in the general solution equation and solve for the constants c1 and c2. This will give us the particular solution to the system.

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