### MAT 132 Spring 2006 Review - Chapter 7, Appendix I, Notes on 2nd order D.E.s

References are to Stewart, Single Variable Calculus - SBU Edition

7.2 Direction fields. Understand how the direction field for the first-order differential equation y' = f(x,y) allows one to "see" the family of solutions [Figures 1-4 page 505]. Be able to sketch the direction field of y' = f(x,y) [Example 1, Exercises 9,10, Exercises 11-14 -choose a 5 by 5 grid of points centered on the initial value]. Given a direction field be able to sketch solutions, and be able to sketch the solution with a given initial value [Figures 7,8 page 506; Example 2; Exercises 1a, 3, 7, 18].
Euler's Method. Given a first-order differential equation y'=f(x,y), an interval [0,T], an initial value y0 and a number N, be able to apply Euler's method with N steps to find approximate values y(T/N), y(2T/N), ... , y(T) for the solution satisfying y(0) = y0 [Example 3 page 509, Example 4 page 510, Exercises 21-24].

7.3 Separable equations. Understand what a separable equation is and be able to solve it by integration, applying the "separate, integrate, solve" method [Examples 1a, 2 page 514]. Understand how the "+C" from one of your integrals turns into the undetermined constant in the general solution, and how an initial value determines what that constant must be [Example 1b, Exercises 9-14].
Mixing problems. Be able to apply the "rate in - rate out" method to convert a mixing problem to a separable differential equation, and then be able to solve the equation [Example 6, problems 35. 37].

7.4 Exponential growth and decay. y' = ky is an important special case of a separable equation. The general solution is y=Cekt, called "exponential growth" if k is positive, "exponential decay" if k is negative. Understand that C = y(0) [Exercises 1,2]. Be able to calculate k from C and one other data point [Exercise 3]. Be able to calculate C and k from two data points [Exercise 4]. Special case: the half-life. Be able to calculate k from the half-life, and vice-versa [Example 3, Exercises 8, 9, 10].
Compound interest. Understand that \$1 invested for one year at annual interest t (in dollars) gives \$1+t if compounded annually, \$(1 + t/2)2 if compounded twice a year \$(1 + t/n)n if compounded n times a year, and \$et if compounded continuously [Example 5 page 531, Exercises 18a, 19a].

Complex Numbers - Appendix I. Understand the identification of the complex number x + iy with the point (x,y) in the plane. Be able to add and multiply complex numbers. [Exercises 1, 2, 3, 4]. Be able to write the complex number x + iy in "polar" form r ei theta. [Box, p.A72, Example 4 p.A73, Exercises 25, 26]. Understand the geometric explanation of multiplication (multiply the r's and add the theta's) [Equato 1, p.A73 and the sentence following].

Second Order Differential Equations (we only study an important special case: linear homogeneous equations with constant coefficients). Review the Notes. Be able to substitute y = elambda x for the unknown function y and to solve for lambda. This method gives two solutions (remember how to get them in the special case where the lambda equation has a double root), say y1 and y2. Be able to write the general solution as c1y1 + c2y2 [Exercises 1, 2 in Notes] and to use initial conditions to calculate c1 and c2 [Exercises 6, 9 in Notes]. The fundamental physical example is a horizontal spring fixed at one end and with a mass of 1 gram at the other. Then, taking y = 0 as the rest position, the position of the mass is given by the solution of y'' + py' + ky = 0 where k is the spring constant and p is the damping constant [Exercises 11-15]