By using variation of parameters, determine the general solution of ty" - 3ty + 4y = t² lnt, t > 0 given that y(t) = t², y(t) = t² Int constitute a fundamental set of solutions of the homogeneous equation. Remark: the corresponding homogeneous differential equation is an Euler equation and fits to item d) in exercise 1.
By using variation of parameters, determine the general solution of ty" - 3ty + 4y = t² lnt, t > 0 given that y(t) = t², y(t) = t² Int constitute a fundamental set of solutions of the homogeneous equation. Remark: the corresponding homogeneous differential equation is an Euler equation and fits to item d) in exercise 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Differential Equations: Variation of Parameters
**Problem Statement:**
By using variation of parameters, determine the general solution of the differential equation
\[
t^2y'' - 3ty' + 4y = t^2 \ln t, \quad t > 0
\]
given that \( y_1(t) = t^2 \), \( y_2(t) = t^2 \ln t \) constitute a fundamental set of solutions of the homogeneous equation.
**Remark:**
The corresponding homogeneous differential equation is an Euler equation and fits to item d) in exercise 1.
### Explanation:
This problem involves solving a second-order non-homogeneous linear differential equation using the method of variation of parameters. Here's a detailed breakdown of the steps involved:
1. **Homogeneous Equation:**
\[
t^2y'' - 3ty' + 4y = 0
\]
Given solutions:
- \( y_1(t) = t^2 \)
- \( y_2(t) = t^2 \ln t \)
2. **Non-Homogeneous Equation:**
\[
t^2y'' - 3ty' + 4y = t^2 \ln t
\]
3. **Variation of Parameters Method:**
To find the particular solution \( y_p(t) \) of the non-homogeneous equation, we assume:
\[
y_p(t) = u_1(t)y_1(t) + u_2(t)y_2(t)
\]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
4. **Determining \( u_1(t) \) and \( u_2(t) \):**
These functions satisfy the following system of equations:
\[
\begin{cases}
u_1'(t)y_1(t) + u_2'(t)y_2(t) = 0 \\
u_1'(t)y_1'(t) + u_2'(t)y_2'(t) = \frac{g(t)}{W(y_1, y_2)}
\end{cases}
\]
where \( W(y_1, y_2) \) is the Wronskian of \( y_1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47a605b-1167-4b41-bc40-883b330bf596%2F2b6c9605-af47-4143-b296-db1038fe9d19%2Fbt2dfgw_processed.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Variation of Parameters
**Problem Statement:**
By using variation of parameters, determine the general solution of the differential equation
\[
t^2y'' - 3ty' + 4y = t^2 \ln t, \quad t > 0
\]
given that \( y_1(t) = t^2 \), \( y_2(t) = t^2 \ln t \) constitute a fundamental set of solutions of the homogeneous equation.
**Remark:**
The corresponding homogeneous differential equation is an Euler equation and fits to item d) in exercise 1.
### Explanation:
This problem involves solving a second-order non-homogeneous linear differential equation using the method of variation of parameters. Here's a detailed breakdown of the steps involved:
1. **Homogeneous Equation:**
\[
t^2y'' - 3ty' + 4y = 0
\]
Given solutions:
- \( y_1(t) = t^2 \)
- \( y_2(t) = t^2 \ln t \)
2. **Non-Homogeneous Equation:**
\[
t^2y'' - 3ty' + 4y = t^2 \ln t
\]
3. **Variation of Parameters Method:**
To find the particular solution \( y_p(t) \) of the non-homogeneous equation, we assume:
\[
y_p(t) = u_1(t)y_1(t) + u_2(t)y_2(t)
\]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
4. **Determining \( u_1(t) \) and \( u_2(t) \):**
These functions satisfy the following system of equations:
\[
\begin{cases}
u_1'(t)y_1(t) + u_2'(t)y_2(t) = 0 \\
u_1'(t)y_1'(t) + u_2'(t)y_2'(t) = \frac{g(t)}{W(y_1, y_2)}
\end{cases}
\]
where \( W(y_1, y_2) \) is the Wronskian of \( y_1
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