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

y[x_]:=Sin[10π Log[x]/Log[2]]

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

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

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

y1[x_]:=Sin[π Log[x]/Log[2]]

y2[x_]:=Sin[2π Log[x]/Log[2]]

Plot[{y1[x],y2[x]},{x,1,2},AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
StyleBox["y", "TI"], "(", 
StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "y (x)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},BaseStyle->FontSize->12,PlotLegends->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"sin", "(", 
FractionBox[
RowBox[{"π", "ln", 
StyleBox["x", "TI"]}], 
RowBox[{"ln", "2"}]], ")"}], TraditionalForm], "errors" -> {}, "input" -> "\\sin \\left( \\frac{\\pi \\ln x}{\\ln 2} \\right)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"sin", "(", 
FractionBox[
RowBox[{"2", "π", "ln", 
StyleBox["x", "TI"]}], 
RowBox[{"ln", "2"}]], ")"}], TraditionalForm], "errors" -> {}, "input" -> "\\sin \\left( \\frac{2 \\pi \\ln x}{\\ln 2} \\right)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}]

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

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

N[3Log[2]/2]

Solve[2π Log[x]/Log[2]==π,x]

Example

Clear[y]

DSolve[{y''[x]+(1+λ) y[x]==0,y[0]==0,y[1]==0},y[x],x]

DSolve[{y''[x]+(2+λ) y[x]==0,y[0]==0,y[1]==0},y[x],x]

y1[x_]:=Sin[π x]

y2[x_]:=Sin[2π x]

Plot[{y1[x],y2[x]},{x,0,1},AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
StyleBox["y", "TI"], "(", 
StyleBox["x", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "y (x)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},BaseStyle->FontSize->12,PlotLegends->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"ϕ", 
RowBox[{"(", 
StyleBox["x", "TI"], ")"}], "==", "sin", 
RowBox[{"(", 
RowBox[{"π", 
StyleBox["x", "TI"]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\phi(x) = \\sin \\left( \\pi x \\right)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{"ψ", 
RowBox[{"(", 
StyleBox["x", "TI"], ")"}], "==", "sin", 
RowBox[{"(", 
RowBox[{"2", "π", 
StyleBox["x", "TI"]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\psi(x) = \\sin \\left( 2 \\pi x \\right)", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}]

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

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