Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Verification of Solution and Graphing Curves
#### Problem Statement
Verify that \( \phi(x) = \frac{4}{1 - ce^x} \), where \( c \) is an arbitrary constant, is a one-parameter family of solutions to
\[ \frac{dy}{dx} = \frac{y^2 - 4y}{4}. \]
Graph the solution curves corresponding to \( c = 0, \pm 1, \pm 2 \) using the same coordinate axes.
#### Solution Verification
1. Substitute \( \phi(x) \) for \( y \) and find the derivative to substitute for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y^2 - 4y}{4} \]
\[ \frac{4}{(1 - ce^x)^2} = \frac{\left( \frac{4}{1 - ce^x} \right)^2 - 4 \left( \frac{4}{1 - ce^x} \right)}{4} \]
2. Simplify the expression:
\[ \frac{4}{(1 - ce^x)^2} = \frac{\frac{16}{(1 - ce^x)^2} - \frac{16}{1 - ce^x}}{4} \]
(Simplify your answer.) The expression on the right can further be simplified by using a common denominator to subtract the fractions in the numerator, resulting in the same expression that was substituted for \( \frac{dy}{dx} \) on the left.
#### Choose the Correct Graph
Choose the graph that shows the family of curves for the specific values of \( c \).
- **Option A**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option B**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option C**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option D**: Graph with curves labeled \( c = 1, 0, -1, 2 \).
Based on the visual inspection of the provided graphs:
- **Option A** depicts a set of hyperbolic-like curves.
- **](https://content.bartleby.com/qna-images/question/c45acf51-cefc-42df-83e4-535a253da58c/bc7566d0-5e0a-4620-bb09-09aaefbfcd9c/61bkpl_processed.jpeg)
Transcribed Image Text:### Verification of Solution and Graphing Curves
#### Problem Statement
Verify that \( \phi(x) = \frac{4}{1 - ce^x} \), where \( c \) is an arbitrary constant, is a one-parameter family of solutions to
\[ \frac{dy}{dx} = \frac{y^2 - 4y}{4}. \]
Graph the solution curves corresponding to \( c = 0, \pm 1, \pm 2 \) using the same coordinate axes.
#### Solution Verification
1. Substitute \( \phi(x) \) for \( y \) and find the derivative to substitute for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y^2 - 4y}{4} \]
\[ \frac{4}{(1 - ce^x)^2} = \frac{\left( \frac{4}{1 - ce^x} \right)^2 - 4 \left( \frac{4}{1 - ce^x} \right)}{4} \]
2. Simplify the expression:
\[ \frac{4}{(1 - ce^x)^2} = \frac{\frac{16}{(1 - ce^x)^2} - \frac{16}{1 - ce^x}}{4} \]
(Simplify your answer.) The expression on the right can further be simplified by using a common denominator to subtract the fractions in the numerator, resulting in the same expression that was substituted for \( \frac{dy}{dx} \) on the left.
#### Choose the Correct Graph
Choose the graph that shows the family of curves for the specific values of \( c \).
- **Option A**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option B**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option C**: Graph with curves labeled \( c = 2, 1, 0, -1, -2 \).
- **Option D**: Graph with curves labeled \( c = 1, 0, -1, 2 \).
Based on the visual inspection of the provided graphs:
- **Option A** depicts a set of hyperbolic-like curves.
- **
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