How to plot ParametricNDSolve

Question

I am trying to solve a differential equation numerically. The code is

H1[z_] = (3 (0.3) (1 + z)^(4) + 
  Exp[a1 u[z]]) (3 - (1/2) (1 + z)^2 (u'[z])^2 )^(-1)
s1 = ParametricNDSolve[{(H1[z])^2 (1 + z)^2 u''[z] - 
   3 (H1[z])^2 (1 + z) u'[z] + a1 Exp[(a1 u[z])] == 0, u'[0] == 1, 
 u[0] == 1}, u, {z, 0, 5}, {a}]

It doesn't plot and evaluate u[z]. I want to plot u[z] for different values of the parameter a1.

Answer 1

First, the parameter in ParametricNDSolve is called a1, not a. Then, ParametricNDSolve returns a rule not a function.

To plot the solutions for different a1, you may write:

H1[z_] = (3  (0.3)  (1 + z)^(4) + 
     Exp[a1  u[z]])  (3 - (1/2)  (1 + z)^2  (u'[z])^2)^(-1);
sol =  u /. 
  ParametricNDSolve[{(H1[z])^2  (1 + z)^2  u''[z] - 
      3  (H1[z])^2  (1 + z)  u'[z] + a1  Exp[(a1  u[z])] == 0, 
    u'[0] == 1, u[0] == 1}, u, {z, 0, 5}, {a1}]

and then e.g.:

funs = Table[so[i][x], {i, 3}];
Plot[funs, {x, 0, 3}]

Answer 2

$Version

(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)

Clear["Global`*"]

H1[z_] = (3   (0.3)   (1 + z)^(4) + 
     Exp[a1   u[z]])   (3 - (1/2)   (1 + z)^2   (u'[z])^2)^(-1);

sol = ParametricNDSolveValue[{(H1[z])^2   (1 + z)^2   u''[z] - 
     3   (H1[z])^2   (1 + z)   u'[z] + a1   Exp[(a1   u[z])] == 0, u'[0] == 1,
    u[0] == 1}, u, {z, 0, 5}, a1]

If sol is used directly in a plot, use Evaluate

Manipulate[
 plt[
  Evaluate[sol[#][z] & /@ {3, 1.5, 1}],
  {z, 0, 5},
  AxesLabel -> (Style[#, 14] & /@ {z, u}),
  PlotLegends ->
   Placed[{3, 1.5, 1}, {.3, .8}]],
 {{plt, Plot, "plot type"}, {Plot, LogPlot}}]