6 Sturm-Liouville Eigenvalue Problems

6 Sturm-Liouville Eigenvalue IntroductionIn the last chapters we have explored the solution of boundary value problemsthat led to trigonometric eigenfunctions. Such functions can be used to repre-sent functions in Fourier series expansions. We would like to generalize someof those techniques in order to solve other boundary value Problems . A class ofproblems to which our previous examples belong and which have eigenfunc-tions with similar properties are the Sturm-Liouville Eigenvalue Problems involve self-adjoint (differential) operators which play an im-portant role in the spectral theory of linear operators and the existence of theeigenfunctions we described in Section These ideas will be introducedin this physics many Problems arise in the form of boundary value problemsinvolving second order ordinary differential equations . For example, we mightwant to solve the equationa2(x)y +a1(x)y +a0(x)y=f(x)( )subject to boundary conditions. We can write such an equation in operatorform by defining the differential operatorL=a2(x)d2dx2+a1(x)ddx+a0(x).

of the λ = −2 case. More specifically, in this case the characteristic equation reduces to r2 = 0. Thus, the general solution of this Cauchy-Euler equation …

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  Equations, Euler, Cauchy, Cauchy euler equation

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