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1. For an autonomous differential equations, it is possible to view all of the solution function gr in terms of "prototypical" graphs. A prototypical solution graph represents an infinite numb other solution graphs. For example, in part (i) below one can view the entire family of func that solve the differential equation in terms of two different prototypical solution graphs separ by an equilibrium solution: one prototypical solution graph is above the t-axis and one is belov t-axis. Each is prototypical because it can stand for all other solution graphs (in its respective re through horizontal translation. Recall the "Making Connections" section of Unit 3. (i) dy dt (ii) = 2y (1 − 1/2) dy dt dy dt = 2y (1-1/2) +3 (iv) dy dt (a) For each differential equation above, draw a phase line and representative graphs of soluti (b) For each differential equation above, explain how your response to number la can be interp

1. For an autonomous differential equations, it is possible to view all of the solution function gr in terms of "prototypical" graphs. A prototypical solution graph represents an infinite numb other solution graphs. For example, in part (i) below one can view the entire family of func that solve the differential equation in terms of two different prototypical solution graphs separ by an equilibrium solution: one prototypical solution graph is above the t-axis and one is belov t-axis. Each is prototypical because it can stand for all other solution graphs (in its respective re through horizontal translation. Recall the "Making Connections" section of Unit 3. (i) dy dt (ii) = 2y (1 − 1/2) dy dt dy dt = 2y (1-1/2) +3 (iv) dy dt (a) For each differential equation above, draw a phase line and representative graphs of soluti (b) For each differential equation above, explain how your response to number la can be interp

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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1. For an autonomous differential equations, it is possible to view all of the solution function graphs in terms of "prototypical" graphs. A prototypical solution graph represents an infinite number of other solution graphs. For example, in part (i) below one can view the entire family of functions that solve the differential equation in terms of two different prototypical solution graphs separated by an equilibrium solution: one prototypical solution graph is above the t-axis and one is below the t-axis. Each is prototypical because it can stand for all other solution graphs (in its respective region) through horizontal translation. Recall the "Making Connections" section of Unit 3. (i) dy dt -Y dy (ii) = 2y (1-2) dy (iii) = 2y (1-1)+3 (iv) du dt (a) For each differential equation above, draw a phase line and representative graphs of solutions. (b) For each differential equation above, explain how your response to number la can be interpreted in terms of prototypical solutions separated by equilibrium solutions.
Transcribed Image Text:1. For an autonomous differential equations, it is possible to view all of the solution function graphs in terms of "prototypical" graphs. A prototypical solution graph represents an infinite number of other solution graphs. For example, in part (i) below one can view the entire family of functions that solve the differential equation in terms of two different prototypical solution graphs separated by an equilibrium solution: one prototypical solution graph is above the t-axis and one is below the t-axis. Each is prototypical because it can stand for all other solution graphs (in its respective region) through horizontal translation. Recall the "Making Connections" section of Unit 3. (i) dy dt -Y dy (ii) = 2y (1-2) dy (iii) = 2y (1-1)+3 (iv) du dt (a) For each differential equation above, draw a phase line and representative graphs of solutions. (b) For each differential equation above, explain how your response to number la can be interpreted in terms of prototypical solutions separated by equilibrium solutions.
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