Solve the given initial-value problem. x²y" + 3xy' = 0, y(1) = 0, y'(1) = 8 y(x) = 4 ,X>0 Use a graphing utility to graph the solution curve. y -4 -2 WebAssign Plot 5 4 X
Solve the given initial-value problem. x²y" + 3xy' = 0, y(1) = 0, y'(1) = 8 y(x) = 4 ,X>0 Use a graphing utility to graph the solution curve. y -4 -2 WebAssign Plot 5 4 X
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 Initial Value Problems in Differential Equations
#### Problem Statement
Solve the given initial-value problem:
\[ x^2 y'' + 3xy' = 0, \quad y(1) = 0, \quad y'(1) = 8 \]
The provided solution to this differential equation is given by:
\[ y(x) = 4 - \frac{4}{t^2}, \quad x > 0 \]
#### Solution Verification
There is an error indicated in the provided solution expression as it points to an incorrect formula.
#### Graphing the Solution
To properly visualize the solution, use a graphing utility to graph the solution curve. Please note that the incorrectly derived formula will lead to incorrect graphing results.
The figure below shows the correct graph of the solution:
**Graph Analysis**
- **Axes**: The x-axis ranges from \(-4\) to \(4\). The y-axis ranges accordingly to provide an appropriate visualization of the solution curve.
- **Solution Curve**: The solution curve, given by \( y(x) \), is plotted in red.
- For values of \( x > 0 \) (as specified in the initial problem conditions), the graph behaves as expected. The curve approaches a finite value as \( x \to 0 \), and increases significantly as \( x \) increases.
**Graph Description**:
The plot depicts:
- The behavior of the solution function \( y(x) \) over the specified domain.
- The red curve shows the trajectory of the solution function, indicating its increase and how it levels off as \( x \) becomes larger.
**Graph Note**: The graph also includes axes labels and tick marks for better comprehension of scales and units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96023f4d-6281-4c81-803e-adfe3e4197d6%2Fd8f2eae7-2312-4f95-9265-10e8c4c40ecf%2Fjvd3dc_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Initial Value Problems in Differential Equations
#### Problem Statement
Solve the given initial-value problem:
\[ x^2 y'' + 3xy' = 0, \quad y(1) = 0, \quad y'(1) = 8 \]
The provided solution to this differential equation is given by:
\[ y(x) = 4 - \frac{4}{t^2}, \quad x > 0 \]
#### Solution Verification
There is an error indicated in the provided solution expression as it points to an incorrect formula.
#### Graphing the Solution
To properly visualize the solution, use a graphing utility to graph the solution curve. Please note that the incorrectly derived formula will lead to incorrect graphing results.
The figure below shows the correct graph of the solution:
**Graph Analysis**
- **Axes**: The x-axis ranges from \(-4\) to \(4\). The y-axis ranges accordingly to provide an appropriate visualization of the solution curve.
- **Solution Curve**: The solution curve, given by \( y(x) \), is plotted in red.
- For values of \( x > 0 \) (as specified in the initial problem conditions), the graph behaves as expected. The curve approaches a finite value as \( x \to 0 \), and increases significantly as \( x \) increases.
**Graph Description**:
The plot depicts:
- The behavior of the solution function \( y(x) \) over the specified domain.
- The red curve shows the trajectory of the solution function, indicating its increase and how it levels off as \( x \) becomes larger.
**Graph Note**: The graph also includes axes labels and tick marks for better comprehension of scales and units.
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