Chapter 3 Second Order Linear Differential Equations

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Chapter 3Second Order Linear Introduction; Basic TerminologyRecall 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 EQUATION:Asecond or-der, Linear differential equationis an equation which can be 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 functionfon the right-hand side is called theforcing functionor thenonhomogeneous term. The term forcing function comes from the applications of Second - Order Equations ;an explanation of the alternative term nonhomogeneous is given Second Order equation which is not Linear is said to on Linear . SetL[y]=y +p(x)y +q(x)y. If we viewLasan operator that transforms a twice differentiable functiony=y(x) into thecontinuous functionL[y(x)] =y (x)+p(x)y (x)+q(x)y(x),39then, for any two twice differentiable functionsy1(x) andy2(x),L[y1(x)+y2(x)] =L[y1(x)] +L[y2(x)]and, for any constantc,L[cy(x)] =cL[y(x)].

homogeneous equations. Homogeneous Equations As defined above, a second order, linear, homogeneous differential equation is an equation that can be written in the form y00 +p(x)y0 +q(x)y = 0 (3) where p and q are continuous functions on some interval I. The Trivial Solution: The first thing to note is that the zero function, y(x)=0

  Equations, Homogeneous, Homogeneous equations