Cv5 - Prvý integrál

<<Matematika`GrCont3D`

?GrCont3D

f (x, y) = (x^2) + (y^2)+x y
je prvý integrál soustavy diferenciálních rovnic
              x' = -x + 2 y
              y' = 2 x + y

Plot3D[x x-y y+x y,{x,-1.2,1.2},{y,-1.2,1.2},
    BoxRatios->{1,1,1}]

GrCont3D[x x-y y+x y,{x,-1.2,1.2},{y,-1.2,1.2},
   {0.0,0.25,0.5,0.75,1.0,-0.5,-1,-1.5},-4,
   PlotRange->{-4,1},BoxRatios->{1,1,1.5}]

f(x,y)=(x^2) + (y^2)+x y 
je prvý integrál soustavy diferenciálních rovnic
              x'=x+2y
               y'=-2x-y

Plot3D[x x+y y+x y,{x,-1.2,1.2},{y,-1.2,1.2},
    BoxRatios->{1,1,1}]

GrCont3D[x x+y y+x y,{x,-1.2,1.2},{y,-1.2,1.2},
   {0.0,0.25,0.5,0.75,1.0,-0.5,-1,-1.5},-4,
   PlotRange->{-4,1},BoxRatios->{1,1,1.5}]

f(x,y)=((x+y Log[y])/y) 
				 je prvý integrál soustavy diferenciálních rovnic
              x'=-x+y
               y'=-y

g1=Plot3D[(x+y Log[y])/y,{x,-1.2,1.2},{y,0.0001,1.2},
    BoxRatios->{1,1,1},PlotPoints->{30,30}]

g2=Plot3D[(x+y Log[Abs[-y]])/y,{x,-1.2,1.2},{y,-1.2,-0.0001},
    BoxRatios->{1,1,1},PlotPoints->{30,30}]

Show[{g1,g2}]

I1=GrCont3D[(x+y Log[y])/y,{x,-1.2,1.2},{y,0.1,1.2},
   {-1,-2,-3,-4,-5,1,2,3,4,5},-20,
   PlotRange->{-20,5},BoxRatios->{1,1,1.5}]

I2=GrCont3D[(x+y Log[-y])/y,{x,-1.2,1.2},{y,-1.2,-0.1},
   {-1,-2,-3,-4,-5,1,2,3,4,5},-20,
   PlotRange->{-20,5},BoxRatios->{1,1,1.5}]

Show[{I1,I2}]

f (x, y) = x y  
je prvý integrál soustavy diferenciálních rovnic
              x' = x
              y' = -y

Plot3D[x y,{x,-2,2},{y,-2,2},
    BoxRatios->{1,1,1}]

GrCont3D[y x,{x,-2,2},{y,-2,2},
   {0.0,0.5,1.0,1.5,-0.5,-1,-2,-3,-4},-10,
   PlotRange->{-10,2},BoxRatios->{1,1,2}]

f (x, y) = y/x 
 je prvý integrál soustavy diferenciálních rovnic
              x' = x
              y' = y

g1=Plot3D[y/x,{x,-2,-0.1},{y,-2,2},
    BoxRatios->{1,1,1}]

g2=Plot3D[y/x,{x,0.1,2},{y,-2,2},
    BoxRatios->{1,1,1}]

Show[{g1,g2}]

I1=GrCont3D[y/x,{x,-2,-0.1},{y,-2,2},
   {0.0,0.5,1.0,-0.5,-1,-2,-3,-4},-10,
   PlotRange->{-10,2},BoxRatios->{1,1,2}]

I2=GrCont3D[y/x,{x,0.1,2.0},{y,-2,2},
   {0.0,0.5,1.0,-0.5,-1,-2,-3,-4},-10,
   PlotRange->{-10,2},BoxRatios->{1,1,2}]

Show[{I1,I2}]

f(x,y)=y-((y^3)/3) -2x+((x^3)/3) 
 je prvý integrál soustavy diferenciálních rovnic
              x'=1-(y^2)
               y'=2-(x^2)

Plot3D[y-y^3/3-2 x+x^3/3,{x,-2,2},{y,-2,2},
    BoxRatios->{1,1,1}]

GrCont3D[y-y^3/3-2 x+x^3/3,{x,-2,2},{y,-2,2},
   {0.0,0.5,1.0,1.5,2,2.5,3,-0.5,-1,-1.5,-2,-2.5,-3},-7,
   PlotRange->{-7,4},BoxRatios->{1,1,2}
   ]

f(x,y)=(1/2) (y^2)-k cos x    je prvý integrál soustavy diferenciálních rovnic
              x'=y
                           y'=-k sin x

k=1;
Plot3D[y*y/2-k Cos[x],{x,-8,8},{y,-2,2},
    BoxRatios->{4,1,1}]

GrCont3D[y*y/2-k Cos[x],{x,-8,8},{y,-2,2},
   {0.0,0.25,0.5,0.75,1.0,-0.5,-1,-1.5},-4,
   PlotRange->{-4,1},BoxRatios->{4,1,1.5}]