8:40 Aa <» Q A CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 14 47. The function y = sin x is an explicit solution of the first-order dy 57. The normal form (5) of an nth-order differen is equivalent to (4) whenever both forms hav same solutions. Make up a first-order differe for which F(x, y, y') = 0 is not equivalent to dy/dx = f(x, y). V1 - y?. Find an interval I of differential equation dx definition. [Hint: I is not the interval (-x, 0).] 48. Discuss why it makes intuitive sense to presume that the linear differential equation y" + 2y' + 4y = 5 sin t has a solution of the form y = A sin t + B cos t, where A and B are constants. Then find specific constants A and B so that y = A sin t + B cos t is a particular solution of the DE. 58. Find a linear second-order differential equatia for which y = c|x + c2x² is a two-parameter f Make sure that your equation is free of the arb cj and c2. In Problems 49 and 50 the given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f(x, y). In each case the relation G(x, y) = 0 implicitly de- fines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off seg- ments, or pieces, on each graph that correspond to graphs of so- lutions. Keep in mind that a solution o must be a function and differentiable. Use the solution curve to estimate an interval I of Qualitative information about a solution y = differential equation can often be obtained fr itself. Before working Problems 59–62, recal significance of the derivatives dy/dx and dy, 59. Consider the differential equation dy/dx = e (a) Explain why a solution of the DE must b function on any interval of the x-axis. definition of each solution o. (b) What are lim dy/dx and lim dy/dx? W 49. 50. y suggest about a solution curve as x→ + (c) Determine an interval over which a solut concave down and an interval over which concave up. (d) Sketch the graph of a solution y = $(x) whose shape is suggest pa 60. Consider the differential equation dy/dx = 5 FIGURE 1.1.6 Graph for FIGURE 1.1.7 Graph for (a) Either by inspection or by the method su Problems 37–40, find a constant solution Problem 49 Problem 50 (b) Using only the differential equation, find y-axis on which a nonconstant solution y 51. The graphs of members of the one-parameter family x + y = 3cxy are called folia of Descartes. Verify that this family is an implicit solution of the first-order differential equation increasing. Find intervals on the y-axis o is decreasing. 61. Consider the differential equation dy/dx = y a and b are positive constants. dy y(y - 2x) x(2y³ – x³) dx (a) Either by inspection or by the method su Problems 37–40, find two constant solut 52. The graph in Figure 1.1.7 is the member of the family of folia in Problem 51 corresponding to c = 1. Discuss: How can the DE in Problem 51 help in finding points on the graph of x + y³ = 3xy where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution o of the DE? Carry out your ideas and compare with your estimates of the intervals in Problem 50. (b) Using only the differential equation, find y-axis on which a nonconstant solution y ing. Find intervals on which y = (x) is (c) Using only the differential equation, exp y = a/2b is the y-coordinate of a point = graph of a nonconstant solution y = 4cx 53. In Examp solutions y = ¢(x) and y = ø2(x) are defined is the open interval (-5, 5). Why can't the interval I of definition be the closed interval [-5, 5]? 7 the largest interval I over which the explicit (d) On the same coordinate axes, sketch the two constant solutions found in part (a) solutions partition the xy-plane into thr each region, sketch the graph of a nonc y = 6(x) whose shape is suggested by parts (b) and (c). 54. In Problem 21 a one-parameter family of solutions of the DE P' = P(1 P) is given. Does any solution curve pass through the point (0, 3)? Through the point (0, 1)? 62. Consider the differential equation y' = y² + 55. Discuss, and illustrate with examples, how to solve differential equations of the forms dy/dx = f(x) and d²y/dx² = f(x). (a) Explain why there exist no constant solu 14 Reader Notebook Bookmarks Flashcards Contents IK ШО