x1 = X2 = Solve the following initial value problem: d (3₂ dt I -15 10 -25 15 x + 10 cos(5t) 12 (10 25(56)), ), Z(0) = (13) 10 sin(5t) 16
x1 = X2 = Solve the following initial value problem: d (3₂ dt I -15 10 -25 15 x + 10 cos(5t) 12 (10 25(56)), ), Z(0) = (13) 10 sin(5t) 16
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
![### Initial Value Problem
#### Solve the following initial value problem:
\[
\frac{d}{dt} \vec{x} =
\begin{pmatrix}
-15 & 10 \\
-25 & 15
\end{pmatrix}
\vec{x} +
\begin{pmatrix}
10 \cos(5t) \\
10 \sin(5t)
\end{pmatrix},
\quad \vec{x}(0) =
\begin{pmatrix}
12 \\
16
\end{pmatrix}
\]
\[
x_1 = \quad \text{[input box]}
\]
\[
x_2 = \quad \text{[input box]}
\]
This problem involves solving a system of first-order differential equations with given initial conditions. The matrix represents the coefficients affecting \(\vec{x}\), and the vector on the right side of the equation represents the non-homogeneous part of the system involving trigonometric functions.
#### Explanation:
- \(\frac{d}{dt} \vec{x}\): Represents the derivative of the vector \(\vec{x}\) with respect to time \(t\).
- \(
\begin{pmatrix}
-15 & 10 \\
-25 & 15
\end{pmatrix}
\): Coefficient matrix influencing the vector \(\vec{x}\).
- \(
\begin{pmatrix}
10 \cos(5t) \\
10 \sin(5t)
\end{pmatrix}
\): Inhomogeneous term that adds a time-dependent component to the system.
- \(\vec{x}(0) =
\begin{pmatrix}
12 \\
16
\end{pmatrix}\): Initial condition specifying the state of the system at \(t = 0\).
#### Goal:
To find \(x_1(t)\) and \(x_2(t)\) that satisfy the given differential equations and initial conditions. These solutions will determine the behavior of the system over time. The solution may involve techniques such as eigenvalue/eigenvector analysis, matrix exponentials, or variation of parameters for systems of differential equations.
Solve for \(x_1(t)\) and \(x_2(t)\) and input your results in the provided boxes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F216015d9-daff-4e98-b3d1-7f3f5202e820%2Ff1ffd845-2c87-4c49-8706-c1b98fa41aa3%2Ftd2dakn_processed.png&w=3840&q=75)
Transcribed Image Text:### Initial Value Problem
#### Solve the following initial value problem:
\[
\frac{d}{dt} \vec{x} =
\begin{pmatrix}
-15 & 10 \\
-25 & 15
\end{pmatrix}
\vec{x} +
\begin{pmatrix}
10 \cos(5t) \\
10 \sin(5t)
\end{pmatrix},
\quad \vec{x}(0) =
\begin{pmatrix}
12 \\
16
\end{pmatrix}
\]
\[
x_1 = \quad \text{[input box]}
\]
\[
x_2 = \quad \text{[input box]}
\]
This problem involves solving a system of first-order differential equations with given initial conditions. The matrix represents the coefficients affecting \(\vec{x}\), and the vector on the right side of the equation represents the non-homogeneous part of the system involving trigonometric functions.
#### Explanation:
- \(\frac{d}{dt} \vec{x}\): Represents the derivative of the vector \(\vec{x}\) with respect to time \(t\).
- \(
\begin{pmatrix}
-15 & 10 \\
-25 & 15
\end{pmatrix}
\): Coefficient matrix influencing the vector \(\vec{x}\).
- \(
\begin{pmatrix}
10 \cos(5t) \\
10 \sin(5t)
\end{pmatrix}
\): Inhomogeneous term that adds a time-dependent component to the system.
- \(\vec{x}(0) =
\begin{pmatrix}
12 \\
16
\end{pmatrix}\): Initial condition specifying the state of the system at \(t = 0\).
#### Goal:
To find \(x_1(t)\) and \(x_2(t)\) that satisfy the given differential equations and initial conditions. These solutions will determine the behavior of the system over time. The solution may involve techniques such as eigenvalue/eigenvector analysis, matrix exponentials, or variation of parameters for systems of differential equations.
Solve for \(x_1(t)\) and \(x_2(t)\) and input your results in the provided boxes.
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