restart;

with(plots):

# Define a custom `rmod` function for periodic switching
rmod := (t, n) -> frac(t * n):

# Create a function for the chessboard pattern that switches periodically
f := (t, p) -> if (floor(4 * t/Pi) + floor(4 * p/Pi)) mod 2 = 0 then 1 else NULL end if:

# Plot the black chessboard squares on the sphere
p1 := plot3d(f, 0 .. 2*Pi, 0 .. Pi, coords = spherical, scaling = constrained, 
             color = "black", grid = [200, 200], style = surface, adaptmesh = false):

# Create a white sphere to make the pattern clearly visible
p2 := sphereplot(0.999, theta = 0 .. 2*Pi, phi = 0 .. Pi, 
                 color = "white", style = surface):

# Combine both plots
plots:-display(p1, p2);

If you follow these axioms in Maple , then it is proved
1 is neutral element for multiplication
0 is neutal element for addition 

Tsol := Re(value(eval(rhs(sols), infinity = 12)))

On school we learned also to bring terms to one side ...:-)

ode := diff(u(x), x, x)/u(x) - diff(u(x), x)^2/u(x)^2 = x^2 - diff(u(x), x)^2/u(x)^2;

# Move all terms to one side
eq := lhs(ode) - rhs(ode);

# Simplify the result
simplified_eq := simplify(eq);

        

@C_R 

Indeed, the cpu load was the same  before and after hibernation.
You must not have created restore points in window 10 to revert to an earlier configuration (impractible, so )?

@C_R 
Amd ryzen 7 5700x- 8-coreprocessor +SDD +16 mb 
Cpu load about average 10 % ? or lower , with or without maple in windows 11 started  , memory 40 % , network 0 % 

Seems there is very little effect from maple on the windows 11 system 
When get things cpu hungry : graphics ? 
Procesverkenner - Sysinternals | Microsoft Learn

@C_R 
I am using Windows 11, so maybe leaving Windows 10 ?

Maybe a unnoticed download on your computer?

@dharr 

Can't the pde be written down more compactly?
This is to avoid clerical errors and to start tracing them, which is a strenuous process 

@salim-barzani 
I think the length of the equation (function rule) is not a problem to include in the table cell. 
Just like a procedure, you can write one block of code using Shift +Enter for the long equation
With F3, you create another prompt.
Of course, the code can also be hidden in the worksheet, showing only a plot with the parameter sliders ( perhaps possible per cell in the table ?). 

This is generated maple code ..hopefully it makes sense :-) 

The two intersections points seems to be outside of the polar grid (roster)? 

A vector field is drawn in the plot of the integral curve of the system of ODEs, but in fact it is a line element field to which no direction can be given.
 

@acer 
" I use 1/100 instead of 0.01, as the extra step(s) to deal with that occlude the main technique in play."
Do i see here 100, or do i miss something ?

@mmcdara 
An error can occur quickly 

expr := exp(x) + exp(2*x) + sin(x);
new_expr := subsindets(expr, exp, f -> ln(f));

               expr := exp(x) + exp(2 x) + sin(x)

Error, type `exp` does not exist

expr := exp(x) + exp(2*x) + sin(x);
new_expr := subsindets(expr, exp(anything), f -> ln(f));

               expr := exp(x) + exp(2 x) + sin(x)

         new_expr := ln(exp(x)) + ln(exp(2 x)) + sin(x)