1166CHAPTER 1717.1Second-Order Linear EquationsAsecond-order linear differential equationhas the formwhere , , , and are continuous functions. We saw in Section 9.1 that equations ofthis type arise in the study of the motion of a spring. In Section 17.3 we will further pur-sue this application as well as the application to electric circuits.In this section we study the case where , for all , in Equation 1. Such equa-tions are calledhomogeneouslinear equations. Thus the form of a second-order linear homo-geneous differential equation isIf for some , Equation 1 isnonhomogeneousand is discussed in Section 17.2.Two basic facts enable us to solve homogeneous linear equations. The first of these saysthat if we know two solutions and of such an equation, then the linear combinationis also a solution.TheoremIf and are both solutions of the linear homogeneous equation and and are any constants, then the functionis also a solution of Equation 2.PROOFSince and are solutions of Equation 2, we haveandTherefore, using the basic rules for differentiation, we haveThus is a solution of Equation 2.P xd2ydx2Q xdydxR x yG xGRQPxG x0P xd2ydx2Q xdydxR x y0xG x0y2y1yc1y1c2y2y2xy1xc2c1y xc1y1xc2y2xy2y1P x y1Q x y1R x y10P x y2Q x y2R x y20P x yQ x yR x yP xc1y1c2y2Q xc1y1c2y2R xc1y1c2y2P xc1y1c2y2Q xc1y1c2y2R xc1y1c2y2c1P x y1Q x y1R x y1c2 P x y2Q x y2R x y2c10c200yc1y1c2y21232
SECOND-ORDER LINEAR EQUATIONS1167The other fact we need is given by the following theorem, which is proved in moreadvanced courses. It says that the general solution is a linear combination of twolinearlyindependentsolutions and This means that neither nor is a constant multiple ofthe other. For instance, the functions and are linearly dependent, butand are linearly independent.TheoremIf and are linearly independent solutions of Equation 2 on aninterval, and is never 0, then the general solution is given bywhere and are arbitrary constants.Theorem 4 is very useful because it says that if we know twoparticular linearly inde-pendent solutions, then we know everysolution.