Use the method of variation of parameters to solve the initial value problem x'=Ax+ f(t), x(a) = x₂ using the following values. 9 -3 36t 1+ 3t 9t ^-[ :-)--[*]-[:]-*-[*** .-*] A = x(0) 3 -1 6t t 1-3t x(t) = f(t): At V
Use the method of variation of parameters to solve the initial value problem x'=Ax+ f(t), x(a) = x₂ using the following values. 9 -3 36t 1+ 3t 9t ^-[ :-)--[*]-[:]-*-[*** .-*] A = x(0) 3 -1 6t t 1-3t x(t) = f(t): At V
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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### Solving the Initial Value Problem using Variation of Parameters
To solve the initial value problem \( \mathbf{x}' = A\mathbf{x} + \mathbf{f}(t) \), \( \mathbf{x}(a) = \mathbf{x}_a \), using the given values, follow these steps.
#### Given Matrices and Initial Conditions:
**Matrix A:**
\[ A = \begin{bmatrix} 9 & -3 \\ 3 & -1 \end{bmatrix} \]
**Function \( \mathbf{f}(t) \):**
\[ \mathbf{f}(t) = \begin{bmatrix} 36t^2 \\ 6t \end{bmatrix} \]
**Initial Condition:**
\[ \mathbf{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
**Exponential Matrix \( e^{At} \):**
\[ e^{At} = \begin{bmatrix} 1 + 3t & -9t \\ t & 1 - 3t \end{bmatrix} \]
#### Objective:
\( \mathbf{x}(t) = \) \(\boxed{\quad}\)
---
### Explanation:
1. **Understand the Problem:**
- The problem involves a system of linear differential equations described by \( \mathbf{x}' = A \mathbf{x} + \mathbf{f}(t) \).
- The method of variation of parameters will be used to find the particular solution of this non-homogeneous equation.
2. **Initial Conditions and Matrices:**
- The initial value provided is \( \mathbf{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \), which means at \( t = 0 \), the state vector \(\mathbf{x}\) is zero.
- The function \( \mathbf{f}(t) \) provides the time-variant input to the system.
- The matrix exponential \( e^{At} \) represents the solution to the homogeneous part of the differential equation.
3. **Steps to Solve:**
- **Step 1:** Solve the homogeneous equation \( \mathbf{x}_h'(t) = A \mathbf{x}_h(t) \) using the given \( e^{At} \).
- **Step 2:** Find the particular solution \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F9f975d06-698a-4188-992c-9b8b61bf8d6e%2Fq4ay40q_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Solving the Initial Value Problem using Variation of Parameters
To solve the initial value problem \( \mathbf{x}' = A\mathbf{x} + \mathbf{f}(t) \), \( \mathbf{x}(a) = \mathbf{x}_a \), using the given values, follow these steps.
#### Given Matrices and Initial Conditions:
**Matrix A:**
\[ A = \begin{bmatrix} 9 & -3 \\ 3 & -1 \end{bmatrix} \]
**Function \( \mathbf{f}(t) \):**
\[ \mathbf{f}(t) = \begin{bmatrix} 36t^2 \\ 6t \end{bmatrix} \]
**Initial Condition:**
\[ \mathbf{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
**Exponential Matrix \( e^{At} \):**
\[ e^{At} = \begin{bmatrix} 1 + 3t & -9t \\ t & 1 - 3t \end{bmatrix} \]
#### Objective:
\( \mathbf{x}(t) = \) \(\boxed{\quad}\)
---
### Explanation:
1. **Understand the Problem:**
- The problem involves a system of linear differential equations described by \( \mathbf{x}' = A \mathbf{x} + \mathbf{f}(t) \).
- The method of variation of parameters will be used to find the particular solution of this non-homogeneous equation.
2. **Initial Conditions and Matrices:**
- The initial value provided is \( \mathbf{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \), which means at \( t = 0 \), the state vector \(\mathbf{x}\) is zero.
- The function \( \mathbf{f}(t) \) provides the time-variant input to the system.
- The matrix exponential \( e^{At} \) represents the solution to the homogeneous part of the differential equation.
3. **Steps to Solve:**
- **Step 1:** Solve the homogeneous equation \( \mathbf{x}_h'(t) = A \mathbf{x}_h(t) \) using the given \( e^{At} \).
- **Step 2:** Find the particular solution \(
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