5389-Chap3 - Chapter 3 Second Order Linear Dierential...

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Chapter 3 Second Order Linear Differential Equations 3.1 Introduction; Basic Terminology Recall that a frst order linear differential equation is an equation which can be written in the Form y ± + p ( x ) y = q ( x ) where p and q are continuous Functions on some interval I . A second order linear differential equation has an analogous Form. A second order, linear differential equation is an equation which can be written in the Form y ±± + p ( x ) y ± + q ( x ) y = f ( x ) (1) where p, q , and f are continuous Functions on some interval I . The Functions p and q are called the coefficients oF the equation; the Function f on the right-hand side is called the forcing function or the nonhomogeneous term . The term “Forcing Function” comes From the applications oF second-order equations; an explanation oF the alternative term “ nonhomogeneous” is given below. A second order equation which is not linear is said to be nonlinear . Remarks on “Linear.” Set L [ y ]= y ±± + p ( x ) y ± + q ( x ) y . IF we view L as an “operator” that transForms a twice differentiable Function y = y ( x ) into the continuous Function L [ y ( x )] = y ±± ( x )+ p ( x ) y ± ( x q ( x ) y ( x ) , then, For any two twice differentiable Functions y 1 ( x ) and y 2 ( x ), L [ y 1 ( x y 2 ( x )] = L [ y 1 ( x )] + L [ y 2 ( x )] 41
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and, for any constant c , L [ cy ( x )] = cL [ y ( x )] . As introduced in Section 2.1, L is a linear operation, speciFcally, a linear differential operator: L : C 2 ( I ) C ( I ) where C 2 ( I ) is the vector space of twice continuously differentiable functions on I and C ( I ) is the vector space of continuous functions on I . ± The Frst 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 (1). Let a be any point on the interval I , and let α and β be any two real numbers. Then the initial-value problem y ±± + p ( x ) y ± + q ( x ) y = f ( x ) ,y ( a )= α, y ± ( a β has a unique solution. A proof of this theorem is beyond the scope of this course. Remark: Chapter 2 gives a method for Fnding the general solution of any Frst order linear equation. In contrast, there is no general method for solving second (or higher) order linear differential equations . There are, however, methods for solving certain special types of second order linear equations and we will consider these in this chapter. ± DEFINITION 1. ( Homogeneous/Nonhomogeneous Equations ) The linear differ- ential equation (1) is homogeneous 1 if the function f on the right side is 0 for all x I . In this case, equation (1) becomes y ±± + p ( x ) y ± + q ( x ) y =0 . (2) Equation (1) is nonhomogeneous if f is not the zero function on I , i.e., (1) is nonhomo- geneous if f ( x ) ± = 0 for some x I . ± ±or reasons which will become clear, almost all of our attention is focused on homoge- neous equations.
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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5389-Chap3 - Chapter 3 Second Order Linear Dierential...

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