### Linear Differential EquationsConsider 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 DiagramsThere 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.

Answered: Consider the vectors x"(t) = (10) and…

Math

Advanced Math

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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Question

### 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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