Solving linear DE systems using fundamental matrix

In summary, solving linear differential equation (DE) systems using the fundamental matrix involves representing the system of equations in matrix form, where the fundamental matrix encapsulates the solutions of the homogeneous system. This matrix is constructed from the linearly independent solutions of the DEs, allowing for the use of matrix exponentiation to find the general solution. The particular solution can be obtained by incorporating initial conditions and using methods such as variation of parameters. This approach provides a systematic way to analyze and solve linear DE systems efficiently.
  • #1
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1715410731470.png


I am confused by the term below. I get all their terms, expect replacing the highlighted term by ##e^{3t}##, does someone please know whether this is yet another typo?

Thanks!
 
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Obviously a typo. And not the first one …
 
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  • #3
Orodruin said:
Obviously a typo. And not the first one …
Thank you for your reply @Orodruin! Do you also please know why they include a comma in the matrix fundamental matrix? It seems a strange use of notation to mean.

Thanks!
 

FAQ: Solving linear DE systems using fundamental matrix

What is a fundamental matrix in the context of linear differential equations?

A fundamental matrix is a matrix solution to a system of linear differential equations that contains all the linearly independent solutions of the system. For a system of first-order linear differential equations, the fundamental matrix can be constructed from the solutions of the system, and it provides a way to express the general solution in terms of initial conditions.

How do you find the fundamental matrix for a linear system of differential equations?

To find the fundamental matrix for a linear system of differential equations, you first need to solve the system to obtain a set of linearly independent solutions. These solutions can then be arranged as columns of a matrix, known as the fundamental matrix. For a system represented as \( \mathbf{y}' = A(t) \mathbf{y} \), where \( A(t) \) is a matrix of coefficients, you can solve the system using methods like the method of undetermined coefficients, variation of parameters, or Laplace transforms, and then construct the fundamental matrix from the solutions.

What is the significance of the determinant of the fundamental matrix?

The determinant of the fundamental matrix, often denoted as \( \Phi(t) \), is significant because it indicates whether the solutions are linearly independent. If the determinant is non-zero for all \( t \) in the interval of interest, it confirms that the solutions are linearly independent and span the solution space of the system. A zero determinant at any point indicates that the solutions are not independent, which can lead to issues in solving the system.

How do you use the fundamental matrix to solve initial value problems?

To solve an initial value problem using the fundamental matrix, you first compute the fundamental matrix \( \Phi(t) \) for the system. If you have an initial condition \( \mathbf{y}(t_0) = \mathbf{y_0} \), you can express the solution as \( \mathbf{y}(t) = \Phi(t) \mathbf{c} \), where \( \mathbf{c} \) is a constant vector determined by the initial condition. You can find \( \mathbf{c} \) by evaluating \( \Phi(t_0) \) and solving the equation \( \Phi(t_0) \mathbf{c} = \mathbf{y_0} \).

Can the fundamental matrix be used for systems with variable coefficients?

Yes, the fundamental matrix can be used for systems with variable coefficients. The process of finding the fundamental matrix remains applicable regardless of whether the coefficients are constant or variable. However, the methods for finding the solutions may differ. For variable coefficient systems, techniques such as reduction of order or series solutions may be employed to derive the fundamental matrix, and once

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