Exam 8: Systems of Linear First-Order Differential Equations

The particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)$ is

(Multiple Choice)

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Question 21

The characteristic equation of $A = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right)$ is

(Multiple Choice)

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Question 22

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

(Multiple Choice)

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Question 23

The eigenvalues of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

(Multiple Choice)

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Question 24

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }$ is

(Multiple Choice)

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Question 25

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right) X$ is

(Multiple Choice)

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Question 26

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

(Multiple Choice)

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Question 27

The Wronskian of the vector functions $X _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }$ and $X _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }$ is

(Multiple Choice)

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Question 28

The Wronskian of the vector functions $\mathbf { X } _ { 1 } = \left( \begin{array} { l } 2 \\1 \\0\end{array} \right) e ^ { t } , \mathbf { X } _ { 2 } = \left( \begin{array} { l } 1 \\3 \\5\end{array} \right) e ^ { 2 t }$ and $X _ { 3 } = \left( \begin{array} { c } - 1 \\5 \\0\end{array} \right) e ^ { 3 t }$ is

(Multiple Choice)

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Question 29

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right) X$ is

(Multiple Choice)

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Question 30

The eigenvalues of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

(Multiple Choice)

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Question 31

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)$ is

(Multiple Choice)

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Question 32

A particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 1 \\t\end{array} \right)$ is

(Multiple Choice)

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Question 33

The characteristic equation for the matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)$ is

(Multiple Choice)

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Question 34

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right) \mathbf { X }$ is

(Multiple Choice)

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Question 35

The eigenvalues of $A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)$ is

(Multiple Choice)

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Question 36

The characteristic equation of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

(Multiple Choice)

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Question 37

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

(Multiple Choice)

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(40)

Question 38

The characteristic equation of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

(Multiple Choice)

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Question 39

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right) \mathbf { X }$ is

(Multiple Choice)

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Question 40

The particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }$ is

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right) X$ is

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

The Wronskian of the vector functions $X _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }$ and $X _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }$ is

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right) X$ is

The eigenvalues of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)$ is

A particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 1 \\t\end{array} \right)$ is

The characteristic equation for the matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right) \mathbf { X }$ is

The eigenvalues of $A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right) \mathbf { X }$ is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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Exam 8: Systems of Linear First-Order Differential Equations

The particular solution of X′=(2321)X+(t1)\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)X′=(22​31​)X+(t1​) is

The characteristic equation of A=(1−32−4)A = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right)A=(12​−3−4​) is

The eigenvalues of A=(2211)A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)A=(21​21​) is

The eigenvalues of A=(2−1−12)A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)A=(2−1​−12​) is

The solution of the system X′=(2211)X\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }X′=(21​21​)X is

The solution of the system X′=(2−1−12)XX ^ { \prime } = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right) XX′=(2−1​−12​)X is

The characteristic equation of A=(−3−221)A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)A=(−32​−21​) is

The Wronskian of the vector functions X1=(1−1)e−2tX _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }X1​=(1−1​)e−2t and X2=(35)e6tX _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }X2​=(35​)e6t is

The solution of the system X′=(−3−221)XX ^ { \prime } = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right) XX′=(−32​−21​)X is

The eigenvalues of A=(−3−221)A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)A=(−32​−21​) is

The characteristic equation of A=(−3−51−1)A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)A=(−31​−5−1​) is

A particular solution of X′=(01−10)X+(1t)\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 1 \\t\end{array} \right)X′=(0−1​10​)X+(1t​) is

The characteristic equation for the matrix A=(30001−1013)A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)A=​300​011​0−13​​ is

The solution of the system X′=(2−222)X\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right) \mathbf { X }X′=(22​−22​)X is

The eigenvalues of A=(1−12−1412−311−2)A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)A=​111​−1221​−14−3−2​​ is

The characteristic equation of A=(2−1−12)A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)A=(2−1​−12​) is

The eigenvalues of A=(2211)A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)A=(21​21​) is

The characteristic equation of A=(2211)A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)A=(21​21​) is

The solution of the system X′=(1−32−4)X\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right) \mathbf { X }X′=(12​−3−4​)X is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

The particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 3 \\2 & 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } t \\1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right) \mathbf { X }$ is

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right) X$ is

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

The Wronskian of the vector functions $X _ { 1 } = \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 2 t }$ and $X _ { 2 } = \left( \begin{array} { l } 3 \\5\end{array} \right) e ^ { 6 t }$ is

The solution of the system $X ^ { \prime } = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right) X$ is

The eigenvalues of $A = \left( \begin{array} { c c } - 3 & - 2 \\2 & 1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { c c } - 3 & - 5 \\1 & - 1\end{array} \right)$ is

A particular solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 0 & 1 \\- 1 & 0\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 1 \\t\end{array} \right)$ is

The characteristic equation for the matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right) \mathbf { X }$ is

The eigenvalues of $A = \left( \begin{array} { c c c } 1 & - 12 & - 14 \\1 & 2 & - 3 \\1 & 1 & - 2\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { c c } 2 & - 1 \\- 1 & 2\end{array} \right)$ is

The eigenvalues of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The characteristic equation of $A = \left( \begin{array} { l l } 2 & 2 \\1 & 1\end{array} \right)$ is

The solution of the system $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 3 \\2 & - 4\end{array} \right) \mathbf { X }$ is