m1[]:=DynamicModule[{X=4,x,y,point={0,-1},splot},
Off[NDSolve::ndsz];
Off[DSolve::bvnul];
splot = StreamPlot[{1,-x/y},{x,-4,4},{y,-X,X}];

LocatorPane[Dynamic[point],
Column[{
Dynamic@("Initial Condition:"<>StringReplace[ToString@point, {"{"->"(","}"->")"}]),
Dynamic@Show[splot,Plot[Evaluate[First[y[x]/. DSolve[{y'[x]==-x/y[x],y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]],{x,-X,X},PlotStyle->Red],ImageSize->400],

Dynamic@("y(x)"<>ToString[First[y[x]/. DSolve[{y'[x]==-x/y[x],y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]/.{x->"x"},TraditionalForm])
}]]

]

m1[]

m2[]:=DynamicModule[{X=4,x,y,point={0,-1},splot},
Off[NDSolve::ndsz];
Off[DSolve::bvnul];
splot = StreamPlot[{1,x+y},{x,-4,4},{y,-X,X}];

LocatorPane[Dynamic[point],
Column[{
Dynamic@("Initial Condition:"<>StringReplace[ToString@point, {"{"->"(","}"->")"}]),
Dynamic@Show[splot,Plot[Evaluate[First[y[x]/. DSolve[{y'[x]==x+y[x],y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]],{x,-X,X},PlotStyle->Red],ImageSize->400],

Dynamic@("y(x)="<>ToString[First[y[x]/. DSolve[{y'[x]==x+y[x],y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]/.{x->"x"},TraditionalForm])
}]]

]

m2[]

m3[]:=DynamicModule[{X=4,x,y,point={0,2},splot},
Off[NDSolve::ndsz];
Off[DSolve::bvnul];
splot = StreamPlot[{1,-y-x^2},{x,-4,4},{y,-X,X}];

LocatorPane[Dynamic[point],
Column[{
Dynamic@("Initial Condition:"<>StringReplace[ToString@point, {"{"->"(","}"->")"}]),
Dynamic@Show[splot,Plot[Evaluate[First[y[x]/. DSolve[{y'[x]==-y[x]-x^2,y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]],{x,-X,X},PlotStyle->Red],ImageSize->400],

Dynamic@("y(x)="<>ToString[First[y[x]/. DSolve[{y'[x]==-y[x]-x^2,y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]/.{x->"x"},TraditionalForm])
}]]

]

m3[]

m4[]:=DynamicModule[{X=4,x,y,point={0,1},splot},
Off[NDSolve::ndsz];
Off[DSolve::bvnul];
Off[Solve::ifun];
splot = StreamPlot[{1,(y+1)*(2-y)/2},{x,-4,4},{y,-X,X}];

LocatorPane[Dynamic[point],
Column[{
Dynamic@("Initial Condition:"<>StringReplace[ToString@point, {"{"->"(","}"->")"}]),
Dynamic@Show[splot,Plot[Evaluate[First[y[x]/. DSolve[{y'[x]==((y[x]+1)*(2-y[x])/2),y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]],{x,-X,X},PlotStyle->Red],ImageSize->400],

Dynamic@("y(x)="<>ToString[First[y[x]/. DSolve[{y'[x]==((y[x]+1)*(2-y[x])/2),y[point[[1]]]==point[[2]]},{y},{x,-X,X}]]/.{x->"x"},TraditionalForm])
}]]

]

m4[]

StreamPlot[{1,-y-x^2},{x,-3,3},{y,-3,3},StreamScale->Full]

Manipulate[StreamDensityPlot[{1,-λ2*y-λ1*x},{x,-3,3},{y,-3,3},StreamScale->Full],{λ1,-3,3},{λ2,-3,3}]

values={-1,0,1};
plots=MapIndexed[Function[{λ,i},StreamDensityPlot[{1,-y-λ*x^2},{x,-3,3},{y,-3,3},StreamPoints->16,StreamScale->0.07,StreamStyle->ColorData["DeepSeaColors"][0.3i[[1]]]]],values]

make3D[GraphicsComplex[pts_,rr___],h_]:=GraphicsComplex[PadRight[pts,{Length@pts,3},h],rr]

make3D[Arrow[pts_,rr___],h_]:=Arrow[PadRight[pts,{Length@pts,3},h],rr]

stackPlots[plots2D:{__Graphics},dh_,o:OptionsPattern[Graphics3D]]:=Graphics3D[MapIndexed[Function[{g,index},g[[1]]/.p:_GraphicsComplex|_Arrow:>make3D[p,dh index[[1]]]],plots2D],o,ImageSize->400]

plotSpacing=5;
stackPlots[plots,plotSpacing,Axes->True,Boxed->False,Ticks->{Automatic,Automatic,MapIndexed[{plotSpacing#2[[1]],Row[{"λ = ",#1}]}&,values]}]