Solution Found!
Consider the differential equation where a and b are
Chapter 1, Problem 57E(choose chapter or problem)
Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\).
Consider the differential equation dy/dx = y(a - by), where a and b are positive constants.
(a) Either by inspection or by the method suggested in Problems 33–36, find two constant solutions of the DE.
(b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on which \(y=\phi(x)\) is decreasing.
(c) Using only the differential equation, explain why y = a/2b is the y-coordinate of a point of inflection of the graph of a non constant solution \(y=\phi(x)\).
(d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the xy-plane into three regions. In each region, sketch the graph of a non constant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c).
Text Transcription:
y=phi(x)
d^2 y / dx^2
Questions & Answers
QUESTION:
Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\).
Consider the differential equation dy/dx = y(a - by), where a and b are positive constants.
(a) Either by inspection or by the method suggested in Problems 33–36, find two constant solutions of the DE.
(b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on which \(y=\phi(x)\) is decreasing.
(c) Using only the differential equation, explain why y = a/2b is the y-coordinate of a point of inflection of the graph of a non constant solution \(y=\phi(x)\).
(d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the xy-plane into three regions. In each region, sketch the graph of a non constant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c).
Text Transcription:
y=phi(x)
d^2 y / dx^2
ANSWER:Step 1 of 5
In this problem, we have to find the solution of an differential equation by using inspection method or some other method.
b)
Then we have to find the increasing interval on the y axis and decreasing interval on the y axis.
c)
We have to explain why be the point of inflection on the graph.
d)
Then we have to stretch the graph of two constant solution found in (a).
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