Math Problem Statement
Solve the IVP: X0 = AX, where A = −2 1 0 −5 0 0 0 3 −2 , and X(0) = −2 −1 2 .
Solution
To solve the initial value problem (IVP) given by , where
and , we need to find the general solution to this system of differential equations and then use the initial condition to find the specific solution.
Step 1: Find the Eigenvalues of
To find the eigenvalues of , we solve the characteristic equation:
where is the identity matrix. Thus,
The determinant is computed as:
[ \begin{vmatrix} -2-\lambda & 1 & 0 \ -5 & -\lambda & 0 \ 0 & 3 & -2-\lambda \end{vmatrix} = (-2-\lambda)[(-\lambda)(-2-\lambda) - (0)(3)]
- 1[-5(0) - (0)(3-\lambda)]
- 0 ]
This simplifies to:
This characteristic polynomial, , factors to:
Thus, the eigenvalues are (with algebraic multiplicity 2), and .
Step 2: Find the Eigenvectors
For :
Solve :
This gives the system of equations:
From , we get . From , with , we get . From , with , we get .
Thus, corresponding to , a single eigenvector .
For :
Solve :
From this, clearly, . The system reduces to:
Setting , the eigenvector associated is .
Step 3: General Solution
The general solution is a combination of matrices and eigenvectors:
With initial condition .
From matched terms, , and . This implies .
Thus, gives the complete solution for .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Differential Equations
Eigenvalues and Eigenvectors
Matrix Exponentiation
Formulas
Matrix Exponential Solution: X(t) = e^(At)X(0)
Characteristic Polynomial: det(A - λI) = 0
Theorems
Eigenvalue Decomposition Theorem
Solution of Linear Differential Equations using Eigenvalues
Suitable Grade Level
Undergraduate - Mathematics or Engineering
Related Recommendation
Solve the IVP X' = AX with Matrix A and Initial Condition X(0)
Solve the Initial Value Problem for X'(t) = AX(t) with Matrix Exponentiation
Solving a Matrix Differential Equation with Initial Condition x(0) = [7, -2]
General Solution for System of Differential Equations with Eigenvalues and Eigenvectors
Solve System of Differential Equations with Matrix Diagonalization