Section 3.4 #6

Eigenvalue condition for λ<0 case:

Plot[{Tanh[z],z},{z,-4,4},PlotStyle->{Green,Purple}]

Eigenvalue condition for λ>0 case:

Plot[{Tan[Sqrt[λ]],Sqrt[λ]},{λ,0,40},PlotStyle->{Green,Purple}]

Find numerical approximation of first positive eigenvalue:

FindRoot[Tan[Sqrt[λ]]==Sqrt[λ],{λ,20}]

λ1=λ/.%

λ1

Look at the corresponding eigenfunction:

y1[x_]=Sin[Sqrt[λ1]x]-Sqrt[λ1]Cos[Sqrt[λ1]x]

Plot[y1[x],{x,0,1}]

Note that this eigenfunction has one zero in the interval  (0,1).  This is consistent with the Sturm-Liouville Theorem since this eigenfunction corresponds to the second eigenvalue in the ordered list of eigenvalues.  (Recall that (Subscript[λ, 0])=0 is an eigenvalue here so (Subscript[λ, 1])≈20.19 is the second eigenvalue and thus (Subscript[y, 1])(x)=sin((Sqrt[(Subscript[λ, 1])])x)-(Sqrt[(Subscript[λ, 1])])cos((Sqrt[(Subscript[λ, 1])])x) is the second eigenfunction.  The SL Theorem tells us that it should have 2-1=1 zero in the interval (0,1).

We can check explicitly that this eigenfunction satisfies the boundary condition y(0)+y'(0)=0.

y1[0]

y1'[0]