The fully dimensional IVP = ODE + IC (Initial Value Problem = Ordinary Differential Equation + Initial Equation) I gave in class on 2020/08/28 is

1/2 m (x’)^2 + m g x = E,     x(0) = 0.

This is Kinetic Energy + Potential Energy = Total Energy (a constant),
for a particle moving on a vertical line with height x.  The mass of the particle is m and g is the acceleration of gravity.

Now, Mathematica gets very touchy if we use E as a variable, so I will use e.  (E is Euler’s constant 2.61828... in Mathematica.  All reserved words in Mathematica are capitalized, so e is OK.)
Aside: Using e here is strange, because a physicist will think “e is the electron charge” and a mathematician with a physics background might  think “e is the negative of the charge of the electron” in that correct but confusing way we math professors have.

This can be written as eqn = 0, where

eqn = 1/2 m xp^2 + m g x - e

I choose to scale the equation by x = 2e/(mg) x-bar and t = (Sqrt[((2e)/(mg^2))])t-bar. The  derivative xp = x’ = dx/dt gets scaled appropriately, and we get the equation

Factor[eqn/.{x -> 2 e/(m g) xbar, xp -> 2 e/(m g) (Sqrt[((m g^2)/(2 e))])xpbar}]

Assume e is not 0, and  “drop the bars”.  The new, non-dimensional equation is
1/2 (((x'))^2) + x = 1/2, 
and the initial condition is still x(0) = 0.
Notice how I set the dimensionless energy = 1/2 so that x’ = 1 or -1  if x = 0.  This is a nicer choice than setting energy = 1 like I did in class.
Look how we can verify that 
x = t - t^2/2 is a solution using Mathematica.

x = t - t^2/2
Simplify[1/2 (D[x,t])^2+ x - 1/2]

This is Nandor Sieben's way of thinking, that differential equations, when written like 1/2 x' + x = 1/2 are solved by expressions (like x = t - t^2/2).
 
 Alternatively, the function x : R to R defined by x (t) = t - t^2/2 is a solution to the ODE written as
1/2 x' (t) + x (t) = 1/2.

In any case, the point is that there are many solutions to this initial value problem.  I graph two solutions here

Plot[{t - t^2/2, -t - t^2/2}, {t, -1, 2}, PlotRange -> {-.2, 1/2}, AxesLabel ->{t, "x"}]

It is worth noting that the following is *not* the graph of a solution to the IVP, since x’ is not defied at t = 0:

Plot[Abs[t] - t^2/2, {t, -1, 2}]

But more interestingly, we can do the magic trick of having the ball suspend in mid-air until time a >  1.  These is a family of solutions to the IVP.  (You might want to verify that  dx/dt is defined and continuous.)  I show the graph of a few members of this family below.  Watch this web site for a possible youtube video!

soln[t_, a_]:= Which[t < 1, t - t^2/2, t < a, 1/2, True, 1/2-(t-a)^2/2]

Plot[{soln[t, 1], soln[t, 2], soln[t, 5]}, {t, 0, 4}, PlotRange ->{-.2, 1/2}, AxesLabel ->{t, "x"}]

The moral of the story is that ODEs of the form 
(4.1)  F(t, x, x’) = 0 
can be very quirky.  Note: (4.1) is the equation numbering in Zachmanoglou and Thoe’s Into to PDE book. We will stick to
(4.2)  x’ = f(t, x).  
This latter, “standard” form also shows that innocent looking ODEs like t x’ + x = t can be very weird at t = 0, since the standard form is
x’ = -x/t + 1.