e4t *(t) = is a solution to the system of linear homogeneous differential equations x = 2x1 + x2 + x3, x2 = x1 + x2 + 2x3, x' = x1 + 2x2 + x3. Find the value of each term in the equation x = 2x1 + x2 + x3 in terms of the variable t.
e4t *(t) = is a solution to the system of linear homogeneous differential equations x = 2x1 + x2 + x3, x2 = x1 + x2 + 2x3, x' = x1 + 2x2 + x3. Find the value of each term in the equation x = 2x1 + x2 + x3 in terms of the variable t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The vector function
\[
\vec{x}(t) = \begin{bmatrix} e^{4t} \\ e^{4t} \\ e^{4t} \end{bmatrix}
\]
is a solution to the system of linear homogeneous differential equations:
\[
x_1' = 2x_1 + x_2 + x_3,
\]
\[
x_2' = x_1 + x_2 + 2x_3,
\]
\[
x_3' = x_1 + 2x_2 + x_3.
\]
**Exercises:**
1. Find the value of each term in the equation \(x_1' = 2x_1 + x_2 + x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]
2. Find the value of each term in the equation \(x_2' = x_1 + x_2 + 2x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]
3. Find the value of each term in the equation \(x_3' = x_1 + 2x_2 + x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F411fdfa3-98d6-4c96-a283-c3e756a59776%2F024262e9-6cf8-40a4-85f4-dde10d4ea855%2Fxk54ctl_processed.png&w=3840&q=75)
Transcribed Image Text:The vector function
\[
\vec{x}(t) = \begin{bmatrix} e^{4t} \\ e^{4t} \\ e^{4t} \end{bmatrix}
\]
is a solution to the system of linear homogeneous differential equations:
\[
x_1' = 2x_1 + x_2 + x_3,
\]
\[
x_2' = x_1 + x_2 + 2x_3,
\]
\[
x_3' = x_1 + 2x_2 + x_3.
\]
**Exercises:**
1. Find the value of each term in the equation \(x_1' = 2x_1 + x_2 + x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]
2. Find the value of each term in the equation \(x_2' = x_1 + x_2 + 2x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]
3. Find the value of each term in the equation \(x_3' = x_1 + 2x_2 + x_3\) in terms of the variable \(t\). (Enter the terms in the order given.)
\[
\boxed{} = \boxed{} + \boxed{} + \boxed{} \quad \text{help (formulas)}
\]
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