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

Plot[-1.5+Sin[π x]+(π-1)x,{x,0.5,7},PlotStyle->Directive[Black],GridLines->{None,{2,6.5,11}+0.738755},Ticks->{{{0.5,"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["a", "TI"], TraditionalForm], "errors" -> {}, "input" -> "$a$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},{6.5,"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["b", "TI"], TraditionalForm], "errors" -> {}, "input" -> "$b$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}},{{-1.5+Sin[π /2]+(π-1)/2,"\!\(\*TemplateBox[<|"boxes" -> FormBox[
SubscriptBox["θ", "0"], TraditionalForm], "errors" -> {}, "input" -> "$\\theta_{0}$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},{2.738755,"\!\(\*TemplateBox[<|"boxes" -> FormBox["π", TraditionalForm], "errors" -> {}, "input" -> "$\\pi$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},{2.738755+4.5,"\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"2", "π"}], TraditionalForm], "errors" -> {}, "input" -> "$2 \\pi$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},{2.738755+9,"\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"3", "π"}], TraditionalForm], "errors" -> {}, "input" -> "$3 \\pi$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}}},AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "$x$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
SubscriptBox["θ", "λ"], "(", 
StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "$\\theta_{\\lambda}(x)$", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},PlotRange->{{0,7},{2.738755-4.5,14}},AxesOrigin->{0,2.738755-4.5},AspectRatio->1/2,BaseStyle->FontSize->12]

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

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

DSolve[{y''[x]+λ y[x]==0,y[0]-y'[0]==0,y[1]+y'[1]==0},y[x],x,Assumptions->{λ>0}]

Plot[2 (Sqrt[λ]) Cos[(Sqrt[λ])]+Sin[(Sqrt[λ])]-λ Sin[(Sqrt[λ])],{λ,0,200}]

Table[FindRoot[2 (Sqrt[λ]) Cos[(Sqrt[λ])]+Sin[(Sqrt[λ])]-λ Sin[(Sqrt[λ])],{λ,x0}],{x0,{1,13,40,90,160}}]

y[λ_][x_]:=(Sqrt[λ]) Cos[x (Sqrt[λ])]+Sin[x (Sqrt[λ])]

Plot[y[161.881][x],{x,0,1},PlotStyle->Directive[Black],AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["y", "TI"], TraditionalForm], "errors" -> {}, "input" -> "y", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},AspectRatio->1/2,BaseStyle->FontSize->12]

Export["example01lambda5.eps",%,"EPS"]

Export["example01lambda5.pdf",%%,"PDF"]

Plot[y[92.76934892142285][x],{x,0,1},PlotStyle->Directive[Black],AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["y", "TI"], TraditionalForm], "errors" -> {}, "input" -> "y", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},AspectRatio->1/2,BaseStyle->FontSize->12]

Export["example01lambda4.eps",%,"EPS"]

Export["example01lambda4.pdf",%%,"PDF"]

Plot[{y[92.76934892142285][x],y[161.88085605098283][x]},{x,0,1},AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["y", "TI"], TraditionalForm], "errors" -> {}, "input" -> "y", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},AspectRatio->1/2,BaseStyle->FontSize->12,PlotLegends->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
SubscriptBox[
StyleBox["X", "TI"], "4"], "(", 
StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "X_{4}(x)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
SubscriptBox[
StyleBox["X", "TI"], "5"], "(", 
StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "X_{5}(x)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}]

Export["example01lambda45.eps",%,"EPS"]

Export["example01lambda45.pdf",%%,"PDF"]