2. By using reduction of order method find another solution y(t) for t²y". 3ty' + 4y = 0, t> 0 given that y₁(t) = t² is a solution. Determine the general solution. Remark: the set {yı(t), y2(t)} is a fundamental set of solutions. -
2. By using reduction of order method find another solution y(t) for t²y". 3ty' + 4y = 0, t> 0 given that y₁(t) = t² is a solution. Determine the general solution. Remark: the set {yı(t), y2(t)} is a fundamental set of solutions. -
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 2: Reduction of Order Method
By using the **reduction of order method**, find another solution \( y_2(t) \) for the differential equation:
\[ t^2 y'' - 3ty' + 4y = 0, \quad t > 0 \]
Given that \( y_1(t) = t^2 \) is a solution, determine the general solution.
**Remark:** The set \(\{y_1(t), y_2(t)\}\) is a fundamental set of solutions.
#### Solution Outline:
1. **Step 1: Assume a form for the second solution.** Since \( y_1(t) \) is known, assume \( y_2(t) = v(t) y_1(t) \) where \( v(t) \) is a function to be determined.
2. **Step 2: Compute derivatives.** Find \( y_2' \) and \( y_2'' \) with the product rule:
- \( y_2(t) = v(t)t^2 \)
- \( y_2' = v'(t)t^2 + 2t v(t) \)
- \( y_2'' = v''(t)t^2 + 4t v'(t) + 2v(t) \)
3. **Step 3: Substitute into the original differential equation.** Plug these into the original differential equation \( t^2 y'' - 3ty' + 4y = 0 \).
4. **Step 4: Simplify the equation.** This will lead to a derivative equation for \( v(t) \).
5. **Step 5: Solve for \( v(t) \).** This involves integrating to find \( v(t) \).
6. **Step 6: Determine \( y_2(t) \) and the general solution.** Combine the solutions to write the general solution.
#### Example Walkthrough:
1. Assume \( y_2(t) = v(t) t^2 \).
2. Calculate:
- \( y_2' = v'(t) t^2 + 2t v(t) \)
- \( y_2'' = v''(t) t^2 + 4t v'(t) + 2 v(t) \)
3. Substitute \( y_2, y_2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47a605b-1167-4b41-bc40-883b330bf596%2F469ca7c2-2e23-43b3-bb29-1a635f7779fe%2F6dgg4f_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Reduction of Order Method
By using the **reduction of order method**, find another solution \( y_2(t) \) for the differential equation:
\[ t^2 y'' - 3ty' + 4y = 0, \quad t > 0 \]
Given that \( y_1(t) = t^2 \) is a solution, determine the general solution.
**Remark:** The set \(\{y_1(t), y_2(t)\}\) is a fundamental set of solutions.
#### Solution Outline:
1. **Step 1: Assume a form for the second solution.** Since \( y_1(t) \) is known, assume \( y_2(t) = v(t) y_1(t) \) where \( v(t) \) is a function to be determined.
2. **Step 2: Compute derivatives.** Find \( y_2' \) and \( y_2'' \) with the product rule:
- \( y_2(t) = v(t)t^2 \)
- \( y_2' = v'(t)t^2 + 2t v(t) \)
- \( y_2'' = v''(t)t^2 + 4t v'(t) + 2v(t) \)
3. **Step 3: Substitute into the original differential equation.** Plug these into the original differential equation \( t^2 y'' - 3ty' + 4y = 0 \).
4. **Step 4: Simplify the equation.** This will lead to a derivative equation for \( v(t) \).
5. **Step 5: Solve for \( v(t) \).** This involves integrating to find \( v(t) \).
6. **Step 6: Determine \( y_2(t) \) and the general solution.** Combine the solutions to write the general solution.
#### Example Walkthrough:
1. Assume \( y_2(t) = v(t) t^2 \).
2. Calculate:
- \( y_2' = v'(t) t^2 + 2t v(t) \)
- \( y_2'' = v''(t) t^2 + 4t v'(t) + 2 v(t) \)
3. Substitute \( y_2, y_2
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