I am trying to understand the shooting method and how to use it. I need it for some more complicated differential equations with boundary conditions at infinity, but I wanted to start with simple things.

I found the following example of BVP in Wiki

$ $ w”(t)=\frac{3}{2}w^2(t)$ $

with $ w(0)=4$ and $ w(1)=0$

The IVP that corresponds to the above

$ $ w”(t)=\frac{3}{2}w^2(t)$ $

with $ w(0)=4$ and $ w'(0)=s$

My implementation in Mathematica.

sols = Map[   NDSolve[{D[w[t], {t, 2}] == 3/2 w[t]^2, w[0] == 4, w[1] == 1}, w, t,      Method -> {"Shooting",        "StartingInitialConditions" -> {w[0] == 4,          w'[0] == #}}] &, {-36, -8, -4, -3, -2, -1, 2, 3}]  {{{w -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}, {{w ->      InterpolatingFunction[{{0., 1.}}, <>]}}} 

So, I get solutions without errors, but when I plot them I only get two discrete lines.

Plot[Evaluate[w[t] /. sols], {t, 0, 10}] 

In Wiki they mention the following:

“was solved for $ s = −1, −2, −3, …, −100$ , and $ F(s) = w(1;s) − 1$ plotted in the first figure. Inspecting the plot of F, we see that there are roots near $ −8$ and $ −36$ . Some trajectories of w(t;s) are shown in the second figure.”

Because of that, I expected that the plots of the solutions to the D.E should be very different since not all the values for $ s$ are roots. And this is my first question. Am I doing something wrong or did I understand something wrong?

The second one: What do they actually mean by $ w(1;s)$ in order to determine which values of $ s$ are the right ones? I don’t know if I should ask this here or in the MathStack.

The relevant wiki link

https://en.wikipedia.org/wiki/Shooting_method