5. Solve the following linear system: dX dt with the initial condition M X (0) = 3 -2 0 3 2 Y
5. Solve the following linear system: dX dt with the initial condition M X (0) = 3 -2 0 3 2 Y
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Solving a Linear System of Differential Equations
### Problem Statement
Solve the following linear system:
\[ \frac{dX}{dt} = \begin{bmatrix} 3 & -2 \\ 0 & 3 \end{bmatrix} X \]
with the initial condition:
\[ X(0) = \begin{bmatrix} -3 \\ 2 \end{bmatrix} \]
### Explanation
To solve this system, you need to:
1. **Find the general solution to the system of differential equations.**
2. **Apply the initial condition to determine the specific solution.**
#### Step 1: Finding the General Solution
For a system of the form:
\[ \frac{dX}{dt} = A X + B \]
where \(A\) is a matrix, and \(B\) is a vector (if \(B\) is zero, we are left with a homogeneous system), we typically start by finding the eigenvalues and eigenvectors of matrix \(A\).
#### Step 2: Applying the Initial Condition
Using the initial condition \(X(0) = \begin{bmatrix} -3 \\ 2 \end{bmatrix}\), solve for any constants in the general solution to find the particular solution that satisfies both the differential equation and the initial condition.
### Graphs/Diagrams
While this problem does not provide direct graphical representation, visualizing the eigenvalues and eigenvectors can be powerful in understanding the system's behavior. In particular, the trajectory of the solution in the phase plane can often be described by the paths dictated by these eigenvectors and corresponding eigenvalues. For a two-dimensional system like this, such trajectories often exhibit behaviors aligned to the eigenvectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa90f48aa-fe1f-4446-8e54-ba82906f66bc%2F38bcb34f-eefe-425a-93a5-356a505ec186%2F5xljnzj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving a Linear System of Differential Equations
### Problem Statement
Solve the following linear system:
\[ \frac{dX}{dt} = \begin{bmatrix} 3 & -2 \\ 0 & 3 \end{bmatrix} X \]
with the initial condition:
\[ X(0) = \begin{bmatrix} -3 \\ 2 \end{bmatrix} \]
### Explanation
To solve this system, you need to:
1. **Find the general solution to the system of differential equations.**
2. **Apply the initial condition to determine the specific solution.**
#### Step 1: Finding the General Solution
For a system of the form:
\[ \frac{dX}{dt} = A X + B \]
where \(A\) is a matrix, and \(B\) is a vector (if \(B\) is zero, we are left with a homogeneous system), we typically start by finding the eigenvalues and eigenvectors of matrix \(A\).
#### Step 2: Applying the Initial Condition
Using the initial condition \(X(0) = \begin{bmatrix} -3 \\ 2 \end{bmatrix}\), solve for any constants in the general solution to find the particular solution that satisfies both the differential equation and the initial condition.
### Graphs/Diagrams
While this problem does not provide direct graphical representation, visualizing the eigenvalues and eigenvectors can be powerful in understanding the system's behavior. In particular, the trajectory of the solution in the phase plane can often be described by the paths dictated by these eigenvectors and corresponding eigenvalues. For a two-dimensional system like this, such trajectories often exhibit behaviors aligned to the eigenvectors.
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