On take (so , , ) with the two boundary conditions y(0)=0

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator

Problem

On [0,1][0,1] take Ly=−y′′\mathcal{L}y=-y'' (so p=1p=1, q=0q=0, w=1w=1) with the two boundary conditions y(0)=0,y(1)=2y′(0).y(0)=0, \qquad y(1)=2y'(0). These are exactly two independent conditions. Decide whether (L,D)(\mathcal{L},\mathcal{D}) is self-adjoint by evaluating the boundary form B(f,g)=[fg′−f′g]01B(f,g)=\big[f g' - f' g\big]_0^1 for the two functions f(x)=x+x2f(x)=x+x^2 and g(x)=x3+xg(x)=x^3+x, both of which satisfy the stated conditions.

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