Use the method of variation of parameters to solve the initial value problem fy" - 2ty' + 2y = ?, y(1) = 4, y'(1) = 3 given that the functions y, =t and y, =ť are linearly independent solutions to the corresponding homogeneous equatio
Use the method of variation of parameters to solve the initial value problem fy" - 2ty' + 2y = ?, y(1) = 4, y'(1) = 3 given that the functions y, =t and y, =ť are linearly independent solutions to the corresponding homogeneous equatio
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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![**Problem Statement:**
Use the method of variation of parameters to solve the initial value problem:
\[ t^2y'' - 2ty' + 2y = t^2, \quad y(1) = 4, \quad y'(1) = 3 \]
Given that the functions \( y_1 = t \) and \( y_2 = t^2 \) are linearly independent solutions to the corresponding homogeneous equation for \( t > 0 \).
**Solution:**
To solve the given differential equation using the method of variation of parameters, we need to find a particular solution based on the given linearly independent solutions \( y_1 \) and \( y_2 \).
### Steps:
1. **General Solution to the Homogeneous Equation:**
The homogeneous equation associated with the problem is:
\[ t^2y'' - 2ty' + 2y = 0 \]
It has the linearly independent solutions:
\[ y_1 = t \]
\[ y_2 = t^2 \]
The general solution to the homogeneous equation is:
\[ y_h = c_1y_1 + c_2y_2 = c_1t + c_2t^2 \]
2. **Particular Solution via Variation of Parameters:**
The particular solution \( y_p \) can be expressed as:
\[ y_p = u_1(t)y_1 + u_2(t)y_2 \]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
3. **Determine \( u_1(t) \) and \( u_2(t) \)**:
By the method of variation of parameters, solve the system of equations:
\[
\begin{align*}
u_1'y_1 + u_2'y_2 &= 0 \\
u_1'y_1' + u_2'y_2' &= \frac{t^2}{t^2}
\end{align*}
\]
Integrate to find \( u_1 \) and \( u_2 \).
4. **Complete Solution:**
The complete solution \( y(t) \) is given by:
\[ y(t) = y_h + y_p \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95e5ca5c-4af5-43e2-9912-788f424e3e60%2F64691eab-1240-4b2c-98d2-c5301da66ec3%2F4hyqcdt_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the method of variation of parameters to solve the initial value problem:
\[ t^2y'' - 2ty' + 2y = t^2, \quad y(1) = 4, \quad y'(1) = 3 \]
Given that the functions \( y_1 = t \) and \( y_2 = t^2 \) are linearly independent solutions to the corresponding homogeneous equation for \( t > 0 \).
**Solution:**
To solve the given differential equation using the method of variation of parameters, we need to find a particular solution based on the given linearly independent solutions \( y_1 \) and \( y_2 \).
### Steps:
1. **General Solution to the Homogeneous Equation:**
The homogeneous equation associated with the problem is:
\[ t^2y'' - 2ty' + 2y = 0 \]
It has the linearly independent solutions:
\[ y_1 = t \]
\[ y_2 = t^2 \]
The general solution to the homogeneous equation is:
\[ y_h = c_1y_1 + c_2y_2 = c_1t + c_2t^2 \]
2. **Particular Solution via Variation of Parameters:**
The particular solution \( y_p \) can be expressed as:
\[ y_p = u_1(t)y_1 + u_2(t)y_2 \]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
3. **Determine \( u_1(t) \) and \( u_2(t) \)**:
By the method of variation of parameters, solve the system of equations:
\[
\begin{align*}
u_1'y_1 + u_2'y_2 &= 0 \\
u_1'y_1' + u_2'y_2' &= \frac{t^2}{t^2}
\end{align*}
\]
Integrate to find \( u_1 \) and \( u_2 \).
4. **Complete Solution:**
The complete solution \( y(t) \) is given by:
\[ y(t) = y_h + y_p \
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