Written from the perspective of the applied mathematician, the latest edition of this bestselling book focuses on the theory and practical applications of Differential Equations to engineering and the sciences. Emphasis is placed on the methods of solution, analysis, and approximation. Use of technology, illustrations, and problem sets help readers develop an intuitive understanding of the material. Historical footnotes trace the development of the discipline and identify outstanding individual contributions. This book builds the foundation for anyone who needs to learn differential equations and then progress to more advanced studies.

initial value a? Let a0 be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of a0 . (b) Solve the initial value problem and find the critical value a0 exactly. (c) Describe the behavior of the solution corresponding to the initial value a0 . ᭤ 23. t y + (t + 1)y = 2te−t , y(1) = a ᭤ 25. Consider the initial value problem ᭤ 24. t y + 2y = (sin t)/t, y + 12 y = 2 cos t, y(−π/2) = a y(0) = −1. Find the coordinates of the first local

mathematical model over wide ranges, whereas this may be very time-consuming or expensive in an experimental setting. Nevertheless, mathematical modeling and experiment or observation are both critically important and have somewhat complementary roles in scientific investigations. Mathematical models s are validated by comparison of their predictions with experimental results. On the other hand, mathematical analyses may suggest the most promising directions to explore experimentally, and may

solution of the difference equation (12) with this value for ρ and the initial condition y0 = 10,000 is given by Eq. (15), that is, yn = (1.01)n (10,000 + 100b) − 100b. (17) The payment b needed to pay off the loan in 4 years is found by setting y48 = 0 and solving for b. This gives b = −100 (1.01)48 = −263.34. (1.01)48 − 1 (18) The total amount paid on the loan is 48 times b or $12,640.32. Of this amount $10,000 is repayment of the principal and the remaining $2640.32 is interest.

Second Order Linear Equations turn, this understanding will assist us in finding the solutions of other problems that we will encounter later. In developing the theory of linear differential equations, it is helpful to introduce a differential operator notation. Let p and q be continuous functions on an open interval I , that is, for α < t < β. The cases α = −∞, or β = ∞, or both, are included. Then, for any function φ that is twice differentiable on I , we define the differential operator L by

α < t < β. Then we must make sure that there is a point in the interval where the Wronskian W of y1 and y2 is nonzero. Under these circumstances y1 and y2 form a fundamental set of solutions and the general solution is y = c1 y1 (t) + c2 y2 (t), where c1 and c2 are arbitrary constants. If initial conditions are prescribed at a point in α < t < β where W = 0, then c1 and c2 can be chosen so as to satisfy these conditions. PROBLEMS In each of Problems 1 through 6 find the Wronskian of the given