Notes on Sturm-Liouville Differential Equations

Notes on Sturm-Liouville Differential EquationsCharles Byrne of Mathematical SciencesUniversity of Massachusetts at LowellLowell, MA 01854, USAA pril 9, 20091 Recalling the Wave EquationThe one-dimensional wave equation is tt(x, t) =c2 xx(x, t),( )wherec >0 is the propagation speed. Separating variables, we seek a solution of theform (x, t) =f(t)y(x). Inserting this into Equation ( ), we getf (t)y(x) =c2f(t)y (x),orf (t)/f(t) =c2y (x)/y(x) = 2,where >0 is the separation constant. We then have the separated differentialequationsf (t) + 2f(t) = 0,( )andy (x) + 2c2y(x) = 0.( )The solutions to Equation ( ) arey(x) = sin( cx).For each arbitrary , the corresponding solution of Equation ( ) isf(t) = sin( t),1orf(t) = cos( t).In the vibrating string problem, the string is fixed at both ends,x= 0 andx=L, sothat (0, t) = (L, t) = 0,for allt. Therefore, we must havey(0) =y(L) = 0, so that the solutions must havethe formy(x) =Amsin( mcx)=Amsin( mLx),where m= cmL, for any positive integerm.

In what follows we shall study the Sturm-Liouville equations, a class of second-order ordinary differential equations that contains, as a special case, the eigenvalue problem in Equation (1.4).

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  Equations, Differential, Sturm, Liouville, On sturm liouville differential equations

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