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Linear Algebra: Solving IVP for X' = AX with Matrix A and Initial Condition
Published on May 29, 2025
Calculus
Solve the initial value problem (IVP) for a system of linear differential equations X' = AX, finding the particular solution given the matrix A and initial condition X(0). The solution involves finding eigenvalues, eigenvectors, and applying the initial condition.
Question
Answer
The problem asks to solve the Initial Value Problem (IVP) for the system of differential equations
The given information is:
Differential equation:
Matrix
Initial condition:
Differential equation:
Matrix
Initial condition:
To solve this IVP, we first need to find the general solution of
Step 1: Find the eigenvalues of
The eigenvalues
The eigenvalues
Setting the determinant to zero:
This gives two eigenvalues:
Step 2: Find the eigenvectors corresponding to each eigenvalue.
For
We need to solve
We need to solve
From the first row,
From the second row,
So, the eigenvector for
From the second row,
So, the eigenvector for
For
We need to solve
We need to solve
From the first row,
Let
So, the eigenvector for
Let
So, the eigenvector for
Step 3: Write the general solution.
The general solution is
The general solution is
Step 4: Use the initial condition to find
We are given
Substitute
We are given
Substitute
Equating this to the given initial condition:
From the second component,
Substitute
Substitute
Step 5: Write the particular solution.
Substitute the values of
Substitute the values of
The final answer is