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Hint: In Problem 35 the transformed equation is a Bernoulli equation. See Problem 23 in Section 2.4. In each of Problems 36 through 37, solve the given initial value problem using the methods of Problems 28 through 35. 36. y'y" = 2, y(0) = 1, y'(0) = 2 37. (1+12) y" +2ty' + 3t-2 = 0, y(1) = 2, y'(1) = -1

Hint: In Problem 35 the transformed equation is a Bernoulli equation. See Problem 23 in Section 2.4. In each of Problems 36 through 37, solve the given initial value problem using the methods of Problems 28 through 35. 36. y'y" = 2, y(0) = 1, y'(0) = 2 37. (1+12) y" +2ty' + 3t-2 = 0, y(1) = 2, y'(1) = -1

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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25. Riccati Equations. The equation dy dt is known as a Riccati25 equation. Suppose that some particular solution y of this equation is known. A more general solution containing one arbitrary constant can be obtained through the substitution Ba 1 v(t) Show that v(t) satisfies the first-order linear equation dv dt Note that v(1) will contain a single arbitrary constant. 26. Verify that the given function is a particular solution of the given Riccati equation. Then use the method of Problem 25 to solve the following Riccati equations: a. y' = 1+1²-2ty + y²; y(t) = t 1 1 b. y' = -√2 = 9₁(1) +92(1)y+q3(t) y² C. dy dt y = y₁(t) + = -(92 +2931) V - 93. 2 t 2 cos² t - sin²t + y² 2 cost +y²; y(t) = t ; yı(t) = sint 27. The propagation of a single action in a large population (for example, drivers turning on headlights at sunset) often depends partly on external circumstances (gathering darkness) and partly on a tendency to imitate others who have already performed the action in question. In this case the proportion y(t) of people who have performed the action can be described 26 by the equation dy/dt = (1-y)(x(t) +by), 29. 30. y" +t(y)² = 0 31. 2t2y" + (y')³ = 2ty', t > 0 Equations with the Independent Variable Missing. Consider second-order differential equations of the form y" = f(y, y'), in which the independent variable t does not appear explicitly. If we let v = y', then we obtain dv/dt = f(y, v). Since the right- hand side of this equation depends on y and v, rather than on t and v, this equation contains too many variables. However, if we think of y as the independent variable, then by the chain rule, dv/dt = (dv/dy) (dy/dt) = v(dv/dy). Hence the original differential equation can be written as v(dv/dy) = f(y, v). Provided that this first-order equation can be solved, we obtain v as a function mation and that of y. A relation between y and t results from solving dy/dt = v(y), which is a separable equation. Again, there are two arbitrary constants in the final result. In each of Problems 32 through 35, use this method to solve the given differential equation. 32. yy" + (y)² = 0 ve the form 33. 33. 34. (28) where x(t) measures the external stimulus and b is the imitation coefficient. a. Observe that equation (28) is a Riccati equation and that y₁ (t) = 1 is one solution. Use the transformation suggested in Problem 25, and find the linear equation satisfied by v(t). b. Find v(t) in the case that x(t) = at, where a is a constant. Leave your answer in the form of an integral. *************** 25 Riccati equations are named for Jacopo Francesco Riccati (1676-1754), a Venetian nobleman, who declined university appointments in Italy, Austria, and Russia to pursue his mathematical studies privately at home. Riccati studied these equations extensively; however, it was Euler (in 1760) who discovered the result stated in this problem. Some Special Second-Order Differential Equations. Second-order differential equations involve the second derivative of the unknown function and have the general form y" = f(t, y, y'). Usually, such equations cannot be solved by methods designed for first-order equations. However, there are two types of second-order equations that can be transformed into first-order equations by a suitable change of variable. The resulting equation can sometimes be solved by the methods presented in this chapter. Problems 28 through 37 deal with these types of equations. 26 See Anatol Rapoport, "Contribution to the Mathematical Theory of Mass Behavior: I. The Propagation of Single Acts," Bulletin of Mathematical Biophysics 14 (1952), pp. 159-169, and John Z. Hearon, "Note on the Theory of Mass Behavior," Bulletin of Mathematical Biophysics 17 (1955), pp. 7-13. Equations with the Dependent Variable Missing. For a second- order differential equation of the form y" = f(t, y'), the substitution V = = y', v' = y" leads to a first-order differential equation of the form v' = f(t, v). If this equation can be solved for v, then y can be obtained by integrating dy/dt = v. Note that one arbitrary constant is obtained in solving the first-order equation for v, and a second is introduced in the integration for y. In each of Problems 28 through 31, use this substitution to solve the given equation. 28. t2y" +2ty' - 1 = 0, t> 0 ty"+y' = 1, t> 0 y" + y = 0 yy"-(y')³ = 0 35. y" + (y)² = 2e-y Hint: In Problem 35 the transformed equation is a Bernoulli equation. See Problem 23 in Section 2.4. In each of Problems 36 through 37, solve the given initial value problem using the methods of Problems 28 through 35. 36. y'y" = 2, y(0) = 1, y'(0) = 2 37. (1+²) y" + 2ty' + 3t2 = 0, y(1) = 2, y'(1) = -1
Transcribed Image Text:25. Riccati Equations. The equation dy dt is known as a Riccati25 equation. Suppose that some particular solution y of this equation is known. A more general solution containing one arbitrary constant can be obtained through the substitution Ba 1 v(t) Show that v(t) satisfies the first-order linear equation dv dt Note that v(1) will contain a single arbitrary constant. 26. Verify that the given function is a particular solution of the given Riccati equation. Then use the method of Problem 25 to solve the following Riccati equations: a. y' = 1+1²-2ty + y²; y(t) = t 1 1 b. y' = -√2 = 9₁(1) +92(1)y+q3(t) y² C. dy dt y = y₁(t) + = -(92 +2931) V - 93. 2 t 2 cos² t - sin²t + y² 2 cost +y²; y(t) = t ; yı(t) = sint 27. The propagation of a single action in a large population (for example, drivers turning on headlights at sunset) often depends partly on external circumstances (gathering darkness) and partly on a tendency to imitate others who have already performed the action in question. In this case the proportion y(t) of people who have performed the action can be described 26 by the equation dy/dt = (1-y)(x(t) +by), 29. 30. y" +t(y)² = 0 31. 2t2y" + (y')³ = 2ty', t > 0 Equations with the Independent Variable Missing. Consider second-order differential equations of the form y" = f(y, y'), in which the independent variable t does not appear explicitly. If we let v = y', then we obtain dv/dt = f(y, v). Since the right- hand side of this equation depends on y and v, rather than on t and v, this equation contains too many variables. However, if we think of y as the independent variable, then by the chain rule, dv/dt = (dv/dy) (dy/dt) = v(dv/dy). Hence the original differential equation can be written as v(dv/dy) = f(y, v). Provided that this first-order equation can be solved, we obtain v as a function mation and that of y. A relation between y and t results from solving dy/dt = v(y), which is a separable equation. Again, there are two arbitrary constants in the final result. In each of Problems 32 through 35, use this method to solve the given differential equation. 32. yy" + (y)² = 0 ve the form 33. 33. 34. (28) where x(t) measures the external stimulus and b is the imitation coefficient. a. Observe that equation (28) is a Riccati equation and that y₁ (t) = 1 is one solution. Use the transformation suggested in Problem 25, and find the linear equation satisfied by v(t). b. Find v(t) in the case that x(t) = at, where a is a constant. Leave your answer in the form of an integral. *************** 25 Riccati equations are named for Jacopo Francesco Riccati (1676-1754), a Venetian nobleman, who declined university appointments in Italy, Austria, and Russia to pursue his mathematical studies privately at home. Riccati studied these equations extensively; however, it was Euler (in 1760) who discovered the result stated in this problem. Some Special Second-Order Differential Equations. Second-order differential equations involve the second derivative of the unknown function and have the general form y" = f(t, y, y'). Usually, such equations cannot be solved by methods designed for first-order equations. However, there are two types of second-order equations that can be transformed into first-order equations by a suitable change of variable. The resulting equation can sometimes be solved by the methods presented in this chapter. Problems 28 through 37 deal with these types of equations. 26 See Anatol Rapoport, "Contribution to the Mathematical Theory of Mass Behavior: I. The Propagation of Single Acts," Bulletin of Mathematical Biophysics 14 (1952), pp. 159-169, and John Z. Hearon, "Note on the Theory of Mass Behavior," Bulletin of Mathematical Biophysics 17 (1955), pp. 7-13. Equations with the Dependent Variable Missing. For a second- order differential equation of the form y" = f(t, y'), the substitution V = = y', v' = y" leads to a first-order differential equation of the form v' = f(t, v). If this equation can be solved for v, then y can be obtained by integrating dy/dt = v. Note that one arbitrary constant is obtained in solving the first-order equation for v, and a second is introduced in the integration for y. In each of Problems 28 through 31, use this substitution to solve the given equation. 28. t2y" +2ty' - 1 = 0, t> 0 ty"+y' = 1, t> 0 y" + y = 0 yy"-(y')³ = 0 35. y" + (y)² = 2e-y Hint: In Problem 35 the transformed equation is a Bernoulli equation. See Problem 23 in Section 2.4. In each of Problems 36 through 37, solve the given initial value problem using the methods of Problems 28 through 35. 36. y'y" = 2, y(0) = 1, y'(0) = 2 37. (1+²) y" + 2ty' + 3t2 = 0, y(1) = 2, y'(1) = -1
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