O Far the two follawing syslems of ODES ; answer the questions: tollowing 1. X(0)- 3 3. -4 - x'(t) = X(0) = 1> Find the eigunvalves of eigenvectors of A. 92Find the quneral salution lignoning initial condition) 3> Find the particular solution (usig the initial eandition)
O Far the two follawing syslems of ODES ; answer the questions: tollowing 1. X(0)- 3 3. -4 - x'(t) = X(0) = 1> Find the eigunvalves of eigenvectors of A. 92Find the quneral salution lignoning initial condition) 3> Find the particular solution (usig the initial eandition)
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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![**For the following systems of ODEs (Ordinary Differential Equations), answer the following questions:**
**(I)**
\[ X'(t) = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix} X, \quad X(0) = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \]
**(II)**
\[ X'(t) = \begin{pmatrix} 5 & -1 \\ 3 & 1 \end{pmatrix} X, \quad X(0) = \begin{pmatrix} 2 \\ -1 \end{pmatrix} \]
1. Find the eigenvalues and eigenvectors of \( A \).
2. Find the general solution (ignoring initial condition).
3. Find the particular solution (using the initial condition).
4. Plot each component of the solution as a function of time.
5. Plot the vector plot (i.e., on the \( x_1, x_2 \)-plane).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1dfec151-558e-4c66-9085-1fd1afee3454%2Fa3812863-ec47-4eba-9301-7dd935b12267%2Ffceonqx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**For the following systems of ODEs (Ordinary Differential Equations), answer the following questions:**
**(I)**
\[ X'(t) = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix} X, \quad X(0) = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \]
**(II)**
\[ X'(t) = \begin{pmatrix} 5 & -1 \\ 3 & 1 \end{pmatrix} X, \quad X(0) = \begin{pmatrix} 2 \\ -1 \end{pmatrix} \]
1. Find the eigenvalues and eigenvectors of \( A \).
2. Find the general solution (ignoring initial condition).
3. Find the particular solution (using the initial condition).
4. Plot each component of the solution as a function of time.
5. Plot the vector plot (i.e., on the \( x_1, x_2 \)-plane).
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