sage: def get_vals(y0,t0,tmax,deltat):
....:     soln = {t0:y0}
....:     y = y0
....:     t = t0
....:     while t < tmax:
....:         dydt = F(y,t)
....:         new_y = y + dydt * deltat
....:         new_t = t + deltat
....:         y = new_y
....:         t = new_t
....:         soln[t] = y
....:     return soln
....:
sage: def F(y,t):
....:     return t*y**2 / (t**4 + 1)
....:
sage: sumval = 0
sage: for i in range(10**5):
....:     sumval += i
....:
sage: for i in range(10**7):
....:     sumval += i
....:
sage: for i in range(10**9):
....:     sumval += i
....:
sage: soln = get_vals(1, 0, 10, 0.001)
sage: sorted(list(soln.items()))[:10]
[(0, 1),
 (0.00100000000000000, 1.00000000000000),
 (0.00200000000000000, 1.00000100000000),
 (0.00300000000000000, 1.00000300000400),
 (0.00400000000000000, 1.00000600002200),
 (0.00500000000000000, 1.00001000007000),
 (0.00600000000000000, 1.00001500017000),
 (0.00700000000000000, 1.00002100034999),
 (0.00800000000000000, 1.00002800064398),
 (0.00900000000000000, 1.00003600109197)]
sage: points(soln.items(), dpi=400, size=1)
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: soln_dict = {}
sage: y0vals = [1 + i*0.1 for i in range(int(9/0.1))]
sage: len(y0vals)
90
sage: soln_dict = {y0: get_vals(y0,0,10,0.001) for y0 in y0vals}
sage: show(sum([points([(k,v) for k,v in tmp.items() if abs(v) >= 0], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
/home/blatm/sage/sage-10.1/src/sage/repl/rich_output/display_manager.py:612: RichReprWarning: Exception in _rich_repr_ while displaying object: Axis limits cannot be NaN or Inf
  warnings.warn(
Graphics object consisting of 90 graphics primitives
sage: show(sum([points([(k,v) for k,v in tmp.items() if (k in RR) and (v in RR)], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
/home/blatm/sage/sage-10.1/src/sage/repl/rich_output/display_manager.py:612: RichReprWarning: Exception in _rich_repr_ while displaying object: Axis limits cannot be NaN or Inf
  warnings.warn(
Graphics object consisting of 90 graphics primitives
sage: show(sum([points([(k,v) for k,v in tmp.items() if (k in RR) and (v in RR) and abs(v) < 100], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
Launched png viewer for Graphics object consisting of 90 graphics primitives
sage: def F(y,t):
....:     return t*cos(y**2) / (t**4 + y)
....:
sage: time soln_dict = {y0: get_vals(y0,0,10,0.001) for y0 in y0vals}
CPU times: user 4.37 s, sys: 54 ms, total: 4.42 s
Wall time: 4.42 s
sage: show(sum([points([(k,v) for k,v in tmp.items() if (k in RR) and (v in RR) and abs(v) < 100], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
Launched png viewer for Graphics object consisting of 90 graphics primitives
sage:
sage: y0vals = [-4.5 + i*0.1 for i in range(int(9/0.1))]
sage: time soln_dict = {y0: get_vals(y0,0,10,0.001) for y0 in y0vals}
CPU times: user 4.18 s, sys: 62 ms, total: 4.24 s
Wall time: 4.24 s
sage: show(sum([points([(k,v) for k,v in tmp.items() if (k in RR) and (v in RR) and abs(v) < 100], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
Launched png viewer for Graphics object consisting of 90 graphics primitives
sage: time soln_dict = {y0: get_vals(y0,0,10,0.0001) for y0 in y0vals}
CPU times: user 38.8 s, sys: 1.13 s, total: 39.9 s
Wall time: 39.9 s
sage: show(sum([points([(k,v) for k,v in tmp.items() if (k in RR) and (v in RR) and abs(v) < 100], dpi=400, size=1, color=hue(float(3*i)/len(soln_dict))) for i,tmp in enumerate(list(soln_dict.values()))]))
Launched png viewer for Graphics object consisting of 90 graphics primitives
sage: