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Solve the given initial-value problem. x²y" + 3xy' = 0, y(1) = 0, y'(1) = 10

Solve the given initial-value problem. x²y" + 3xy' = 0, y(1) = 0, y'(1) = 10

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Solving Initial-Value Problems with Differential Equations ### Problem Statement: Solve the given initial-value problem. \[ x^2 y'' + 3x y' = 0, \quad y(1) = 0, \quad y'(1) = 10 \] \[ y(x) = \underline{\hspace{40px}}, \quad x > 0 \] ### Graphical Solution: Use a graphing utility to graph the solution curve. Below are four possible solutions represented by different graphs. #### Graph Analysis: 1. **First Graph (Top-Left Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -20 to 5. - **Curve**: The solution curve is red and starts close to \( x \approx 0.5 \), increases steeply from \( y \approx -20 \) to \( y \approx 5 \), then levels off as x increases. 2. **Second Graph (Top-Right Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -20 to 5. - **Curve**: The solution curve is red, starting steeply from \( x \approx 0 \) and \( y \approx -20 \) and leveling off to \( y \approx 0 \) as x reaches around 4. 3. **Third Graph (Bottom-Left Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -5 to 20. - **Curve**: The solution curve is red, beginning from \( y = 20 \) at \( x \approx 0 \), then it sharply decreases to \( y \approx 0 \) as \( x \) approaches 5. 4. **Fourth Graph (Bottom-Right Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -5 to 20. - **Curve**: The red solution curve starts from \( y = 20 \) when \( x \approx 0 \), decreasing steadily and continuing to decrease below the x-axis as \( x \) increases. ### Conclusion: From these graphs, analyze which solution curve best fits the given initial conditions and behavior
Transcribed Image Text:## Solving Initial-Value Problems with Differential Equations ### Problem Statement: Solve the given initial-value problem. \[ x^2 y'' + 3x y' = 0, \quad y(1) = 0, \quad y'(1) = 10 \] \[ y(x) = \underline{\hspace{40px}}, \quad x > 0 \] ### Graphical Solution: Use a graphing utility to graph the solution curve. Below are four possible solutions represented by different graphs. #### Graph Analysis: 1. **First Graph (Top-Left Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -20 to 5. - **Curve**: The solution curve is red and starts close to \( x \approx 0.5 \), increases steeply from \( y \approx -20 \) to \( y \approx 5 \), then levels off as x increases. 2. **Second Graph (Top-Right Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -20 to 5. - **Curve**: The solution curve is red, starting steeply from \( x \approx 0 \) and \( y \approx -20 \) and leveling off to \( y \approx 0 \) as x reaches around 4. 3. **Third Graph (Bottom-Left Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -5 to 20. - **Curve**: The solution curve is red, beginning from \( y = 20 \) at \( x \approx 0 \), then it sharply decreases to \( y \approx 0 \) as \( x \) approaches 5. 4. **Fourth Graph (Bottom-Right Quadrant)** - **Axes**: The x-axis ranges from -4 to 5; the y-axis ranges from -5 to 20. - **Curve**: The red solution curve starts from \( y = 20 \) when \( x \approx 0 \), decreasing steadily and continuing to decrease below the x-axis as \( x \) increases. ### Conclusion: From these graphs, analyze which solution curve best fits the given initial conditions and behavior
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