x+ *(0) = Form the complementary solution to the homogeneous equation. x.(t) = a1 + a2 help (formulas) help (matrices) Construct a particular solution by assuming the form x,(t) = de?' + bt + ĉ and solving for the undetermined constant vectors a, b, and č. i,(1) = help (formulas) help (matrices) Form the general solution x(t) = x.(t) + x„(t) and impose the initial condition to obtain the solution of the initial value problem. x1(t) x2(t) help (formulas) help (matrices)
x+ *(0) = Form the complementary solution to the homogeneous equation. x.(t) = a1 + a2 help (formulas) help (matrices) Construct a particular solution by assuming the form x,(t) = de?' + bt + ĉ and solving for the undetermined constant vectors a, b, and č. i,(1) = help (formulas) help (matrices) Form the general solution x(t) = x.(t) + x„(t) and impose the initial condition to obtain the solution of the initial value problem. x1(t) x2(t) help (formulas) help (matrices)
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
![**Consider the initial value problem**
\[ \vec{x}' = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \vec{x} + \begin{bmatrix} t \\ e^{2t} \end{bmatrix}, \quad \vec{x}(0) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \]
---
**Form the complementary solution to the homogeneous equation.**
\[ \vec{x}_c(t) = \alpha_1 \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} + \alpha_2 \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]
---
**Construct a particular solution by assuming the form** \( \vec{x}_p(t) = \vec{a}e^{2t} + \vec{b}t + \vec{c} \) **and solving for the undetermined constant vectors** \(\vec{a}\), \(\vec{b}\), **and** \(\vec{c}\).
\[ \vec{x}_p(t) = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]
---
**Form the general solution** \( \vec{x}(t) = \vec{x}_c(t) + \vec{x}_p(t) \) **and impose the initial condition to obtain the solution of the initial value problem.**
\[ \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F411fdfa3-98d6-4c96-a283-c3e756a59776%2Fcfb8738d-555d-4cfa-a208-c3b6ddce4f91%2Fxx8tczj_processed.png&w=3840&q=75)
Transcribed Image Text:**Consider the initial value problem**
\[ \vec{x}' = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \vec{x} + \begin{bmatrix} t \\ e^{2t} \end{bmatrix}, \quad \vec{x}(0) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \]
---
**Form the complementary solution to the homogeneous equation.**
\[ \vec{x}_c(t) = \alpha_1 \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} + \alpha_2 \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]
---
**Construct a particular solution by assuming the form** \( \vec{x}_p(t) = \vec{a}e^{2t} + \vec{b}t + \vec{c} \) **and solving for the undetermined constant vectors** \(\vec{a}\), \(\vec{b}\), **and** \(\vec{c}\).
\[ \vec{x}_p(t) = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]
---
**Form the general solution** \( \vec{x}(t) = \vec{x}_c(t) + \vec{x}_p(t) \) **and impose the initial condition to obtain the solution of the initial value problem.**
\[ \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix} \]
\[ \text{help (formulas)} \quad \text{help (matrices)} \]
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