Match the differential equation with the appropriate integral curves, if it shown.
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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Transcribed Image Text:**Title: Differential Equation and Integral Curve Matching**
**Objective:**
Match the given differential equation with its corresponding integral curves.
**Instructions:**
Select the subfigure that appropriately matches each differential equation described below. Note that not all equations may have a suitable integral curve represented.
**Subfigures:**
- **Subfigure A:**
- A plot showing directional field lines and integral curves within the plane.
- **Subfigure B:**
- A plot featuring directional field lines and integral curves intersecting at various points, indicating changes in direction.
- **Subfigure C:**
- A plot displaying smooth directional field lines and integral curves, showcasing a stable direction.
- **Subfigure D:**
- A plot illustrating directional field lines and integral curves with prominent peaks and valleys, highlighting complex interactions.
**Equations:**
(a) \( y' = -y(2-y) \) matches with subfigure **B**
(b) \( y' = -(y-2)^2(y+2) \) matches with subfigure **C**
(c) \( y' = y^2(y+2)(y-2) \) matches with subfigure **D**
**Key Notes:**
- The subfigures depict different behaviors of differential equations in the context of directional fields and potential integral curves.
- Understanding the relation between differential equations and their graphical representations is crucial for comprehension of dynamic systems in a mathematical context.
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