Second Order Linear Differential Equations

CHAPTER 3 Second Order Linear Differential Introduction; Basic Terminology and ResultsAny Second Order differential equation can be written asF(x,y,y ,y )=0 This chapter is concerned with special yet very important Second Order Equations , namelylinear that a first Order Linear differential equation is an equation which can be writtenin the formy +p(x)y=q(x)wherepandqare continuous functions on some intervalI. A Second Order , lineardifferential equation has an analogous Order Linear differential equationis an equation which canbe written in the formy +p(x)y +q(x)y=f(x)(1)wherep, q, andfare continuous functions on some functionspandqare called thecoefficientsof the equation; the functionfonthe right-hand side is called theforcing functionor thenonhomogeneous term. The term forcing function comes from applications of Second - Order Linear Equations ; the description nonhomogeneous is given Second Order equation which is not Linear is said to (a)y 5y +6y= 3cos2x.

The first 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 y00 +p(x)y0 +q(x)y= f(x),y(a)=α, y0(a)=β

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