Homogeneous differential equation solutions difficulties

Questions
, ,
1

Hi,
Every one.
I appreciate if someone can help me.

I’ve been taking difficulties with solutions of homogeneous differential equations. I’m not sure with this solution.

I think that there is a problem with Vector Plot and equations family.

Open in Wolfram Cloud ↗
2

I am afraid I do not understand your problem, but what do you think about this?

Define a tangent-vector

tvec[u_, f_] := Module[{},
  abl = D[f, x];
  v1 = {u, f /. x -> u};
  v2 = {u + .1, (f /. x -> u) + .1 (abl /. x -> u)};
  {v1, v2}
  ]

and then plot these tangent-vectors with the solutions of your diff.eq.

Show[
Graphics[
{Arrow /@ Table[tvec[a, funcionesh[[2]]], {a, 0, 2, .25}],
Arrow /@ Table[tvec[a, funcionesh[[4]]], {a, 0, 2, .25}]}],
famh,
Axes -> True,
AxesOrigin -> {0, 0}
    ]

OK, that ist not yet really nice, but with a bit of playing around with the parameters it could be acceptable.

3

Try:

  funcionesh = Table[y[x, c1], {c1, 0, 10, 1/5}];
  famh = Plot[funcionesh, {x, -5, 5}, PlotRange -> {-30, 0}, 
    AxesLabel -> {"x", "y"}, AxesStyle -> Black, PlotRange -> All]
  campoh = VectorPlot[{1, (x - y)/(x + y)}, {x, -5, 6}, {y, -30, 20}, 
    Axes -> True, AxesStyle -> Black, Axes -> Black]
  Show[{famh, campoh}]

Regards M.I.

4

Just to put into words what is going on.

You defined the vector field in VectorPlot to be {(x-y)/(x+y), x}.

The vector field should be {1, (x-y)/(x+y)}.

General remark: A vector field of the form (v_x, v_y) represents the system of differential equations dx/dt = v_x, dy/dt = v_y.

Thus a vector field of the form (1, f(x, y)) represents the system dx/dt = 1, dy/dt = f(x, y)). Since dy/dx = \big(dy/dt\big) \big/ \big(dx/dt\big), this system is equivalent to dy/dx = f(x,y).

Gratuitous remarks:

The integration parameter shows up as E^(2 C[1]), which can be negative if C[1] is complex. This is inconvenient, perhaps. We can replace C[1] -> Log[c1]/2 to get simply c1 as a parameter, which now give negative term inside Sqrt[] whenever c1 is negative.

A characteristic of homogeneous DEs is that the integral curves cross a radial line at the same angle. Since in recent versions of WL in VectorPlot, many vectors lie on radial lines at angles of 0º, 30º, 60º, etc., this is easy to show.

solhomo2 = solhomo /. C[1] -> Log[c1]/2  (* put parameter in a convenient form *)
(*  {{y[x] -> -x - Sqrt[c1 + 2 x^2]}, {y[x] -> -x + Sqrt[c1 + 2 x^2]}}  *)

funcionesh = Flatten@Table[y[x] /. solhomo2,
  {c1, 2 Abs[#] # &@Range[-3, 3]}]; (* c1 = ±2 times squares *)
Show[
  VectorPlot[{1, (x - y)/(x + y)}, {x, -5, 5}, {y, -5, 5},
    (*Axes->True,AxesStyle->Black,Axes->Black,*)
    Prolog -> { (* omit axes, add radial lines *)
      Thin, Black,
      Table[
        InfiniteLine[{0, 0}, {Cos[t], Sin[t]}], {t, 0, 5 Pi/6, Pi/6}],
      Dashed,
      Table[
        InfiniteLine[{0, 0}, {Cos[t], Sin[t]}], {t, Pi/12, Pi, Pi/6}]}
   ],
  Plot[funcionesh, {x, -10, 10}]
]

Untitled
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5

Thanks for your help.

6

Thanks for your help.

7

Don’t worry. It’s o.k. By the way, I’m not working in this area.

Sorry.