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Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Equations

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Fundamentals Of Differential Equations And Boundary Value Problems
Pages: 90 - 151
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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102 Questions for Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Equations

  1. An object of mass 2 kg is released from rest from a platform 30 m above the water and allowed to fall under the influence of gravity. After the object strikes the water, it begins to sink with gravity pulling down and a buoyancy force pushing up. Assume that the force of gravity is constant, no change in momentum occurs on impact with the water, the buoyancy force is 1/2 the weight (weight = mg), and the force due to air resistance or water resistance is proportional to the velocity, with proportionality constant b1= 10 N-sec/m in the air and b2= 100 N-sec/m in the water. Find the equation of motion of the object. What is the velocity of the object 1 min after it is released?

    Found on Page 116

  2. In Example 1, we solved for the velocity of the object as a function of time (equation (5)). In some cases, it is useful to have an expression, independent of t, that relates vand x.Find this relation for the motion in Example 1. [Hint: Lettingv(t)=V(xt), thendvdt=(dvdx)V]

    Found on Page 116

  3. The solution to the initial value problem\({\bf{y' = }}\frac{{\bf{2}}}{{{{\bf{x}}^{\bf{4}}}}}{\bf{ - }}{{\bf{y}}^{\bf{2}}}{\bf{,y(1) = - 0}}{\bf{.414}}\), crosses the x-axis at a point in the interval \(\left[ {{\bf{1,2}}} \right]\).By experimenting with the fourth-order Runge–Kutta subroutine, determine this point to two decimal places

    Found on Page 139

  4. A shell of mass 2 kg is shot upward with an initial velocity of 200 m/sec. The magnitude of the force on the shell due to air resistance is |v|/20. When will the shell reach its maximum height above the ground? What is the maximum height

    Found on Page 116

  5. By experimenting with the fourth-order Runge-Kutta subroutine, find the maximum value over the interval \(\left[ {{\bf{1,2}}} \right]\)of the solution to the initial value problem\({\bf{y' = }}\frac{{{\bf{1}}{\bf{.8}}}}{{{{\bf{x}}^{\bf{4}}}}}{\bf{ - }}{{\bf{y}}^{\bf{2}}}{\bf{,y(1) = - 1}}\) . Where does this maximum occur? Give your answers to two decimal places.

    Found on Page 139

  6. The solution to the initial value problem \(\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ = }}{{\bf{y}}^{\bf{2}}}{\bf{ - 2}}{{\bf{e}}^{\bf{x}}}{\bf{y + }}{{\bf{e}}^{{\bf{2x}}}}{\bf{ + }}{{\bf{e}}^{\bf{x}}}{\bf{,y(0) = 3}}\)has a vertical asymptote (“blows up”) at some point in the interval\(\left[ {{\bf{0,2}}} \right]\). By experimenting with the fourth-order Runge–Kutta subroutine, determine this point to two decimal places.

    Found on Page 139

  7. When the velocity v of an object is very large, the magnitude of the force due to air resistance is proportional to v2 with the force acting in opposition to the motion of the object. A shell of mass 3 kg is shot upward from the ground with an initial velocity of 500 m/sec. If the magnitude of the force due to air resistance is 0.1v2, when will the shell reach its maximum height above the ground? What is the maximum height?

    Found on Page 116

  8. Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problem\({\bf{y' = ycosx,y(0) = 1}}\) , at \({\bf{x = \pi }}\). For a tolerance of \({\bf{\varepsilon = 0}}{\bf{.01}}\) use a stopping procedure based on the absolute error.

    Found on Page 140

  9. Use the fourth-order Runge–Kutta subroutine with h= 0.1 to approximate the solution to\({\bf{y' = cos}}\;{\bf{5y - x,y(0) = 0}}\),at the points x= 0, 0.1, 0.2, . . ., 3.0. Use your answers to make a rough sketch of the solution on\(\left[ {{\bf{0,3}}} \right]\).

    Found on Page 140

  10. Use the fourth-order Runge–Kutta subroutine with h= 0.1 to approximate the solution to\({\bf{y' = 3cos(y - 5x),y(0) = 0}}\) , at the points x= 0, 0.1, 0.2, . . ., 4.0. Use your answers to make a rough sketch of the solution on [0, 4].

    Found on Page 140

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