Families of Solutions for ODEs

Manipulate[StreamPlot[{1,f@@{x,y,aa,bb,cc,dd}},{x,-3,3},{y,-3,3},Axes->True,AxesStyle->Arrowheads[{0.0,0.05}],AxesLabel->{x,y},Frame->False,AxesOrigin->{-3,-3},PlotRange->{{-3,3},{-3,3}},ImageSize->{400,400}],{{f,#3 ( #1^#4)+#5( #2^#6)&,Style["(dy/dx) =",12]},{#3 ( #1^#4)+#5( #2^#6)&->TraditionalForm[a (x^b) + c (y^d)],#3 ( #1^#4)( #2^#5)&->TraditionalForm[a (x^b)(y^c)],#3 Sin[#4 #1]+#5 Cos[#6  #2]&->TraditionalForm[a Sin[b x] + c Cos[d y]],#3 Cos[ #1^#4]+#5 Sin[ #2^#6]&->TraditionalForm[a Cos[(x^b)] + c Sin[(y^d)]]},PopupMenu},{{aa,1,"a"},-5,5,1,Appearance->"Labeled",ImageSize->Tiny},
{{bb,1,"b"},-5,5,1,Appearance->"Labeled",ImageSize->Tiny},
{{cc,1,"c"},-5,5,1,Appearance->"Labeled",ImageSize->Tiny},{{dd,1,"d"},-5,5,1,Appearance->"Labeled",ImageSize->Tiny,Enabled->If[f==(#3 ( #1^#4)( #2^#5)&),False,True]},ControlPlacement->Left,SynchronousUpdating->False]

CAPTION

This Demonstration explores families of solutions of an ordinary differential equation (ODE) of the form ((dy)/(dx))=f(x,y), which are plotted using the vector field (1,f(x,y) ). For each choice of f you can see how the solutions depend on the value of the parameters; in some cases going from negative to positive values causes a significant change in the behavior of the solutions.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Families of Solutions for ODEs" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/FamiliesOfSolutionsForODEs/

Contributed by: Gosia Konwerska

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