Consider the vectors x"(t) = (10) and x"() - 8t
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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![### Linear Differential Equations
Consider the vectors \( x^{(1)}(t) = \begin{pmatrix} t \\ 10 \end{pmatrix} \) and \( x^{(2)}(t) = \begin{pmatrix} t^2 \\ 8t \end{pmatrix} \).
#### (a) Compute the Wronskian of \( x^{(1)} \) and \( x^{(2)} \).
\[ W = \begin{vmatrix} t & t^2 \\ 10 & 8t \end{vmatrix} = \boxed{} \]
#### (b) In what intervals are \( x^{(1)} \) and \( x^{(2)} \) linearly independent?
\[ D = ( \boxed{} , \boxed{} ) \cup ( \boxed{} , \boxed{} ) \]
#### (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by \( x^{(1)} \) and \( x^{(2)} \)?
\[ \text{Choose one} \]
\[ \text{of the coefficients of the ODE in standard form must be discontinuous at } t_0 = \boxed{} \]
#### (d) Find the system of equations \( \mathbf{x}' = P(t)\mathbf{x} \).
\[ P(t) = \begin{pmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{pmatrix} \]
### Explanation of the Graphs and Diagrams
There are no graphical elements present in this problem. The given mathematical expressions involve vectors, determinants for the Wronskian, and a matrix for the system of differential equations.
This problem requires the computation of the Wronskian to determine the linear independence of two vector functions and the subsequent formulation of a system of differential equations. The intervals of linear independence are determined based on the values of \( t \) which can be identified from the Wronskian's computation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2Fbac77a70-57f9-4bcf-b5a4-09375aac79a2%2Ftq0o4tx_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Differential Equations
Consider the vectors \( x^{(1)}(t) = \begin{pmatrix} t \\ 10 \end{pmatrix} \) and \( x^{(2)}(t) = \begin{pmatrix} t^2 \\ 8t \end{pmatrix} \).
#### (a) Compute the Wronskian of \( x^{(1)} \) and \( x^{(2)} \).
\[ W = \begin{vmatrix} t & t^2 \\ 10 & 8t \end{vmatrix} = \boxed{} \]
#### (b) In what intervals are \( x^{(1)} \) and \( x^{(2)} \) linearly independent?
\[ D = ( \boxed{} , \boxed{} ) \cup ( \boxed{} , \boxed{} ) \]
#### (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by \( x^{(1)} \) and \( x^{(2)} \)?
\[ \text{Choose one} \]
\[ \text{of the coefficients of the ODE in standard form must be discontinuous at } t_0 = \boxed{} \]
#### (d) Find the system of equations \( \mathbf{x}' = P(t)\mathbf{x} \).
\[ P(t) = \begin{pmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{pmatrix} \]
### Explanation of the Graphs and Diagrams
There are no graphical elements present in this problem. The given mathematical expressions involve vectors, determinants for the Wronskian, and a matrix for the system of differential equations.
This problem requires the computation of the Wronskian to determine the linear independence of two vector functions and the subsequent formulation of a system of differential equations. The intervals of linear independence are determined based on the values of \( t \) which can be identified from the Wronskian's computation.
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