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dy_ vy – v,V + y² dx

dy_ vy – v,V + y² dx

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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I need to do the exercices 28,29,30,31 from page 104-105 from the book "A first course in differential equations with modeling applications" by Deniss G. Zill

If someone can help me, i would really appreciate it 

28. Old Man River ... In Figure 3.2.8(a) suppose that the y-axis and the dashed vertical line x = 1 represent, re- spectively, the straight west and east bheaches of a river that is 1 mile wide. The river flows northward with a velocity v, where Iv,l = v, mi/h is a constant. A man enters the current at the point (1, 0) on the east shore and swims in a direction and rate relative to the river given by the vector v, where the speed Iv,l = v, mi/h is a constant. The man wants to reach the west beach exactly at (0, 0) and so swims in such a manner that keeps his velocity vector v, always directed toward the point (0, 0). Use Figure 3.2.8(b) as an aid in showing that a mathematical model for the path of the swimmer in the river is dy _ vy – v,V² + y? dx [Hint: The velocity v of the swimmer along the path or curve shown in Figure 3.2.8 is the resultant v = v, + v,. Resolve v, and v, into components in the x- and swimmer west east beach beach current (0, 0) (1, 0) * (a) (x(1), y(t)) y() (0, 0) x(1) (1, 0) * (b)
Transcribed Image Text:28. Old Man River ... In Figure 3.2.8(a) suppose that the y-axis and the dashed vertical line x = 1 represent, re- spectively, the straight west and east bheaches of a river that is 1 mile wide. The river flows northward with a velocity v, where Iv,l = v, mi/h is a constant. A man enters the current at the point (1, 0) on the east shore and swims in a direction and rate relative to the river given by the vector v, where the speed Iv,l = v, mi/h is a constant. The man wants to reach the west beach exactly at (0, 0) and so swims in such a manner that keeps his velocity vector v, always directed toward the point (0, 0). Use Figure 3.2.8(b) as an aid in showing that a mathematical model for the path of the swimmer in the river is dy _ vy – v,V² + y? dx [Hint: The velocity v of the swimmer along the path or curve shown in Figure 3.2.8 is the resultant v = v, + v,. Resolve v, and v, into components in the x- and swimmer west east beach beach current (0, 0) (1, 0) * (a) (x(1), y(t)) y() (0, 0) x(1) (1, 0) * (b)
y-directions. If x = x(1), y = y(t) are parametric equa- tions of the swimmer's path, then v = (dx/dt, dy/dt).] 29. (a) Solve the DE in Problem 28 subject to y(1) = 0. For convenience let k = v,/v,- (b) Determine the values of v, for which the swimmer will reach the point (0, 0) by examining_ lim y(x) in the cases k = 1, k>1, and 0<k< 1. 30. Old Man River Keeps Moving . Suppose the man in Problem 28 again enters the current at (1, 0) but this time decides to swim so that his velocity vector v, is always directed toward the west beach. Assume that the speed Iv,l = v, mi/h is a constant. Show that a mathe- matical model for the path of the swimmer in the river is now dy dx 31. The current speed v, of a straight river such as that in Problem 28 is usually not a constant. Rather, an approxi- mation to the current speed (measured in miles per hour) could be a function such as v,(x) = 30x(1 – x), 0 x 1, whose values are small at the shores (in this case, v,(0) = 0 and v,(1) = 0) and largest in the middle of the river. Solve the DE in Problem 30 subject to y(1) = 0, where v, = 2 mi/h and v,(x) is as given. When the swim- mer makes it across the river, how far will he have to walk along the beach to reach the point (0, 0)?
Transcribed Image Text:y-directions. If x = x(1), y = y(t) are parametric equa- tions of the swimmer's path, then v = (dx/dt, dy/dt).] 29. (a) Solve the DE in Problem 28 subject to y(1) = 0. For convenience let k = v,/v,- (b) Determine the values of v, for which the swimmer will reach the point (0, 0) by examining_ lim y(x) in the cases k = 1, k>1, and 0<k< 1. 30. Old Man River Keeps Moving . Suppose the man in Problem 28 again enters the current at (1, 0) but this time decides to swim so that his velocity vector v, is always directed toward the west beach. Assume that the speed Iv,l = v, mi/h is a constant. Show that a mathe- matical model for the path of the swimmer in the river is now dy dx 31. The current speed v, of a straight river such as that in Problem 28 is usually not a constant. Rather, an approxi- mation to the current speed (measured in miles per hour) could be a function such as v,(x) = 30x(1 – x), 0 x 1, whose values are small at the shores (in this case, v,(0) = 0 and v,(1) = 0) and largest in the middle of the river. Solve the DE in Problem 30 subject to y(1) = 0, where v, = 2 mi/h and v,(x) is as given. When the swim- mer makes it across the river, how far will he have to walk along the beach to reach the point (0, 0)?
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