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Match the differential equation with its direction field. y' = 6-y // 0-2 VIII -1 1-Q.41 -Q.2\ 11 1 / 0.4/ 1 1 1 --|||||||||| 1 2 - 0A / 0-2 O //// -0.4 \ / / / / / / / -1 11.P / \ \ \ / 0 \ || / // \ \ I / / / / / / //// | | | / / L F \ \ \\ 1 2

Match the differential equation with its direction field. y' = 6-y // 0-2 VIII -1 1-Q.41 -Q.2\ 11 1 / 0.4/ 1 1 1 --|||||||||| 1 2 - 0A / 0-2 O //// -0.4 \ / / / / / / / -1 11.P / \ \ \ / 0 \ || / // \ \ I / / / / / / //// | | | / / L F \ \ \\ 1 2

Advanced Engineering Mathematics
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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### Provide Reasons for Your Answer: --- **Question:** Given the differential equation \( y' = 6 - y \), provide reasons why each of the following statements may be true or false. 1. **\( y' = 6 - y = 0 \) on the lines \( x = 0 \) and \( y = 6 \).** - This statement claims that the derivative \( y' \) is zero, indicating a horizontal slope, along the line \( x = 0 \) for all \( y \), and specifically on the line \( y = 6 \). 2. **The slopes at each point are independent of \( x \), so the slopes are the same along each line parallel to the x-axis. Note that for \( y = 6 \), \( y' = 0 \).** - This statement suggests that \( y' \) does not depend on \( x \) but only on \( y \), indicating that the slope of the function is constant along lines parallel to the x-axis. It reinforces this by noting that at \( y = 6 \), the slope is zero. 3. **\( y' = 6 - y = 0 \) on the line \( y = -x + 1/6 \), and \( y' = -1 \) on the line \( y = -x \).** - This statement equates \( y' \) to zero on the specific line \( y = -x + 1/6 \), and assigns a specific value of \(-1\) to \( y' \) on the line \( y = -x \). 4. **The slopes at each point are independent of \( y \), so the slopes are the same along each line parallel to the y-axis. Note that for \( y = 6 \), \( y' = 0 \).** - This statement suggests that \( y' \) does not depend on \( y \) but only on \( x \), implying that the slope of the function is constant along lines parallel to the y-axis. It also notes the zero slope for \( y = 6 \). 5. **\( y' = 6 - y = 0 \) on the lines \( x = 0 \) and \( y = 0 \), and \( y' > 0 \) for \( 0 < x < \
Transcribed Image Text:### Provide Reasons for Your Answer: --- **Question:** Given the differential equation \( y' = 6 - y \), provide reasons why each of the following statements may be true or false. 1. **\( y' = 6 - y = 0 \) on the lines \( x = 0 \) and \( y = 6 \).** - This statement claims that the derivative \( y' \) is zero, indicating a horizontal slope, along the line \( x = 0 \) for all \( y \), and specifically on the line \( y = 6 \). 2. **The slopes at each point are independent of \( x \), so the slopes are the same along each line parallel to the x-axis. Note that for \( y = 6 \), \( y' = 0 \).** - This statement suggests that \( y' \) does not depend on \( x \) but only on \( y \), indicating that the slope of the function is constant along lines parallel to the x-axis. It reinforces this by noting that at \( y = 6 \), the slope is zero. 3. **\( y' = 6 - y = 0 \) on the line \( y = -x + 1/6 \), and \( y' = -1 \) on the line \( y = -x \).** - This statement equates \( y' \) to zero on the specific line \( y = -x + 1/6 \), and assigns a specific value of \(-1\) to \( y' \) on the line \( y = -x \). 4. **The slopes at each point are independent of \( y \), so the slopes are the same along each line parallel to the y-axis. Note that for \( y = 6 \), \( y' = 0 \).** - This statement suggests that \( y' \) does not depend on \( y \) but only on \( x \), implying that the slope of the function is constant along lines parallel to the y-axis. It also notes the zero slope for \( y = 6 \). 5. **\( y' = 6 - y = 0 \) on the lines \( x = 0 \) and \( y = 0 \), and \( y' > 0 \) for \( 0 < x < \
**Differential Equation and Directional Field Matching** **Problem Statement:** Match the differential equation with its direction field. **Given Differential Equation:** \[ y' = 6 - y \] **Direction Fields:** Four direction fields are provided, each represented in a separate graph. 1. **Top-Left Graph:** - Range for x-axis: -2 to 2 - Range for y-axis: 0 to 12 - The slope field lines converge toward \( y = 6 \). 2. **Top-Right Graph:** - Range for x-axis: -2 to 2 - Range for y-axis: 0 to 12 - The slope field lines converge toward \( y = 0 \). 3. **Bottom-Left Graph:** - Range for x-axis: -0.4 to 0.4 - Range for y-axis: -0.4 to 0.4 - The slope field lines diverge symmetrically around the origin. 4. **Bottom-Right Graph:** - Range for x-axis: -0.4 to 0.4 - Range for y-axis: -0.4 to 0.4 - The slope field lines converge towards \( y = -0.4 \). **Explanation of Graphs:** Each graph shows a grid with the x-axis and y-axis. The arrows indicate the direction of the solution curve of the differential equation at each point in the plane. Each arrow's slope matches the value of \( y' \) at that point. **Objective:** Determine which direction field corresponds to the differential equation \( y' = 6 - y \). **Steps to Match:** 1. Evaluate the equilibrium point(s) by setting \( y' = 0 \): \[ 6 - y = 0 \implies y = 6 \] This means the solution should tend toward \( y = 6 \). 2. Analyze the behavior of the slopes: - If \( y < 6 \), then \( y' > 0 \) (slopes are positive). - If \( y > 6 \), then \( y' < 0 \) (slopes are negative). From this analysis, the **top-left graph** is the correct match, as the direction field shows the slopes converging
Transcribed Image Text:**Differential Equation and Directional Field Matching** **Problem Statement:** Match the differential equation with its direction field. **Given Differential Equation:** \[ y' = 6 - y \] **Direction Fields:** Four direction fields are provided, each represented in a separate graph. 1. **Top-Left Graph:** - Range for x-axis: -2 to 2 - Range for y-axis: 0 to 12 - The slope field lines converge toward \( y = 6 \). 2. **Top-Right Graph:** - Range for x-axis: -2 to 2 - Range for y-axis: 0 to 12 - The slope field lines converge toward \( y = 0 \). 3. **Bottom-Left Graph:** - Range for x-axis: -0.4 to 0.4 - Range for y-axis: -0.4 to 0.4 - The slope field lines diverge symmetrically around the origin. 4. **Bottom-Right Graph:** - Range for x-axis: -0.4 to 0.4 - Range for y-axis: -0.4 to 0.4 - The slope field lines converge towards \( y = -0.4 \). **Explanation of Graphs:** Each graph shows a grid with the x-axis and y-axis. The arrows indicate the direction of the solution curve of the differential equation at each point in the plane. Each arrow's slope matches the value of \( y' \) at that point. **Objective:** Determine which direction field corresponds to the differential equation \( y' = 6 - y \). **Steps to Match:** 1. Evaluate the equilibrium point(s) by setting \( y' = 0 \): \[ 6 - y = 0 \implies y = 6 \] This means the solution should tend toward \( y = 6 \). 2. Analyze the behavior of the slopes: - If \( y < 6 \), then \( y' > 0 \) (slopes are positive). - If \( y > 6 \), then \( y' < 0 \) (slopes are negative). From this analysis, the **top-left graph** is the correct match, as the direction field shows the slopes converging
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