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# Note for MATH 3321 at UH

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Date Created: 02/06/15

CHAPTER 3 Second Order Linear Differential Equations 31 Introduction Basic Terminology and Results Any second order differential equation can be written as Fm 247 27 24 0 This chapter is concerned with special yet very important second order equations namely linear equations Recall that a rst order linear differential equation is an equation which can be written in the form 1 2905 6196 where p and q are continuous functions on some interval 1 A second order linear differential equation has an analogous form DEFINITION 1 A second order linear di erential equation is an equation which can be written in the form 24 py qy 1W 1 where p q and f are continuous functions on some interval I The functions p and q are called the coe cients of the equation the function f on the right hand side is called the forcing function or the nonhomogeneous term The term forcing function77 comes from applications of second order linear equations the description nonhomogeneous is given below A second order equation which is not linear is said to be nonlinear Examples a y 7 53 6y 3 cos 2x Here pz 75 qz 6 x 3 cos 2x are continuous functions on foo 00 b z2 y 7 2x 3 2y 0 This equation is linear because it can be written in the form 1 as 29 iii y 0 where pz 2m qz 2m2 x 0 are continuous on any interval that does not contain z O For example we could take I 0 oo 63 c y zyzy 7 y3 e is a nonlinear equation this equation cannot be written in the form I Remarks on Linear Intuitively a second order differential equation is linear if y appears in the equation with exponent 1 only and if either or both of y and 3 appear in the equation then they do so with exponent 1 only Also there are no so called cross product77 terms yy y y y y In this sense it is easy to see that the equations in a and b are linear and the equation in c is nonlinear Set My y pzy If we View L as an operator that transforms a twice differentiable function y into the continuous function Llyxl 24 Nell90 610596 then for any two twice differentiable functions y1z and y2m Lly1 y2xl 24196 y2xl ply1 y2xl qly1 y2l arm y z Plyl yam qzy1z yam We pyl qy1 24390 py 6100 we Lly1ml Lly2l and for any constant 0 WWW l0yxl plcyxl qlcyl cy pl0y wl 6610596 ellTm Nell90 qyl cLlyl Therefore as introduced in Section 21 L is a linear dz erentz39al operator This is the real reason that equation 1 is said to be a linear differential equation I The rst thing we need to know is that an initial value problem has a solution and that it is unique THEOREM 1 Existence and Uniqueness Theorem Given the second order linear equation Let a be any point on the interval I and let Oz and B be any two real numbers Then the initial value problem 24 2496 2 6100 y f7 Ma at Ma B has a unique solution 64 As before a proof of this theorem is beyond the scope of this course Remark We can solve any first order linear differential equation Section 2 1 gives a method for nding the general solution of any first order linear equation In contrast there is no general method for solving second or higher order linear di erential equations There are however methods for solving certain special types of second order linear equations and we shall study these in this chapter Extensions of these methods to higher order linear equations will be given later I DEFINITION 2 The linear differential equation 1 is homogeneous 1 if the function f on the right side of the equation is 0 for all z E I In this case equation 1 becomes 24 2496 2 6196 y 0 2 Equation 1 is nonhomogeneous if f is not the zero function on I ie 1 is nonhomo geneous if fz 7 0 for some z E I As you will see in the work which follows almost all of our attention will be focused on homogeneous equations 1This use of the term homogeneous is completely different from its use to categorize the rst order equation y fxy in Exercises 22 65

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