SetDirectory[$HomeDirectory<>"/Documents/handouts/math467/GeneralizedFourierSeries"]

{c1+c2-β1 k(c1-c2)==0,c1 Exp[k]+ c2 Exp[-k]+β2 k(c1 Exp[k]-c2 Exp[-k])==0}

Solve[%,{c1,c2}]

{c2-β1 c1==0,c1+c2+β2 c2==0}

Solve[%,{c1,c2}]

X[x_]:=c1 Cos[k x]+c2 Sin[k x]

{X[0]-β1 X'[0]==0,X[1]+β2 X'[1]==0}

Solve[%,{c1,c2}]

%7[[2]]/.c1->β1 k c2

Plot[{Tan[k],2k/((k^2)-1)},{k,0,13},AxesLabel->{"k",""},Exclusions->{k==1,Cos[k]==0},ExclusionsStyle->Directive[Thin,Black,Dashed],PlotPoints->50,PlotStyle->{Automatic,Thick},PlotRange->{-5,5},Ticks->{Table[n π/2,{n,1,8,1}],Automatic},PlotLegends->"Expressions"]

Export["Thirdspectrum.eps",%,"EPS"]

Export["Thirdspectrum.pdf",%%,"PDF"]

Table[FindRoot[Tan[k]==2k/((k^2)-1),{k,(n-1)π}],{n,1,6,1}]

(k^2)/.Rest[%]