1、Ch 3.1: Second Order Linear Homogeneous Equations with Constant Coefficients,A second order ordinary differential equation has the general form where f is some given function. This equation is said to be linear if f is linear in y and y:Otherwise the equation is said to be nonlinear. A second order
2、linear equation often appears asIf G(t) = 0 for all t, then the equation is called homogeneous. Otherwise the equation is nonhomogeneous.,Homogeneous Equations, Initial Values,In Sections 3.6 and 3.7, we will see that once a solution to a homogeneous equation is found, then it is possible to solve t
3、he corresponding nonhomogeneous equation, or at least express the solution in terms of an integral. The focus of this chapter is thus on homogeneous equations; and in particular, those with constant coefficients:We will examine the variable coefficient case in Chapter 5. Initial conditions typically
4、 take the formThus solution passes through (t0, y0), and slope of solution at (t0, y0) is equal to y0.,Example 1: Infinitely Many Solutions (1 of 3),Consider the second order linear differential equationTwo solutions of this equation areOther solutions includeBased on these observations, we see that
5、 there are infinitely many solutions of the formIt will be shown in Section 3.2 that all solutions of the differential equation above can be expressed in this form.,Example 1: Initial Conditions (2 of 3),Now consider the following initial value problem for our equation:We have found a general soluti
6、on of the formUsing the initial equations, Thus,Example 1: Solution Graphs (3 of 3),Our initial value problem and solution areGraphs of this solution are given below. The graph on the right suggests that both initial conditions are satisfied.,Characteristic Equation,To solve the 2nd order equation w
7、ith constant coefficients,we begin by assuming a solution of the form y = ert. Substituting this into the differential equation, we obtainSimplifying, and hence This last equation is called the characteristic equation of the differential equation. We then solve for r by factoring or using quadratic
8、formula.,General Solution,Using the quadratic formula on the characteristic equationwe obtain two solutions, r1 and r2. There are three possible results: The roots r1, r2 are real and r1 r2. The roots r1, r2 are real and r1 = r2. The roots r1, r2 are complex. In this section, we will assume r1, r2 a
9、re real and r1 r2. In this case, the general solution has the form,Initial Conditions,For the initial value problemwe use the general solution together with the initial conditions to find c1 and c2. That is,Since we are assuming r1 r2, it follows that a solution of the form y = ert to the above init
10、ial value problem will always exist, for any set of initial conditions.,Example 2,Consider the initial value problemAssuming exponential soln leads to characteristic equation:Factoring yields two solutions, r1 = -4 and r2 = 3 The general solution has the formUsing the initial conditions:Thus,Example 4: Initial Value Problem (1 of 2),Consider the initial value problem ThenFactoring yields two solutions, r1 = -2 and r2 = -3 The general solution has the formUsing initial conditions:Thus,Example 4: Find Maximum Value (2 of 2),Find the maximum value attained by the solution.,
