@vv Motivated by your comment, I produced and posted some graphics.    Cheers!

Motivated by vv 1627 's comments, I produced some graphics which may be interest.

This one shows the Earth-Sun system as viewed from the "above".  The "Spring", "Summer", etc., labels pertain to the northern hemisphere.

This one is the same picture shown at an angle:

And here is an animation of Earth going around the sun.  It may take some time to load if you have a slow connection.

You will find the code that produced these graphics here: daylight-orbit.mw

@fereydoon_shekofte I am glad that it was of some help.

I found that the calculations could be improved.  I have posted a new version of the maple worksheet as an addendum to my original message.

Cheers!

A phase-plane plot makes sense for autonomous differential equations, that is, differential equation whose coefficients do not depend on time.  The cos(wt) term makes your equation nonautonomous.  A phase-plane plot, even if you manage to produce one, conveys no useful information.  You should rethink your question.

Preben, I an unable to execute this code.  I run into an error in computing J1a.  I am puzzled, because the error is due to the Change(J1,s=-t) which seems to be rather innocuous.  I have attached a worksheet to show what I am getting.

mw.mw

Rather than replying individually, I want to issue this collective Thank You for the very insightful solutions and comments from which I have learned quite a bit.

A car's forward acceleration is produced through its engine, which uses fuel  The reverse acceleratiom is produced by the breaks, which use no fuel.  Your formula for the energy consumption, that is integral F_acc(s) ds, should be changed to

int(max(0,F_acc(s)), s=0..d);

That corrects the problem's formulation.  I haven't thought about its solution.

@ The help page for "fsolve,details" recommends specifying an interval with fsolve.  Thus, we do:

sys := {6*exp(x+y)/(1+exp(x+y))+exp(2*y)/(1+exp(2*y)) = 2,
            5*exp(2*x)/(1+exp(2*x))+2*exp(x+y)/(1+exp(x+y)) = 1};
fsolve(sys, {x=-2..0,y=-1..1});

and we get

I suppose that you expect the solution to be x = 2*A - 5.

Now, you tell me, what does it mean to subtract a number from a vector?

What equation are you trying to solve?

@erik10  Let's say we replace the first equation by:

0.44 - a < PA^2 + 2*PA*PB < 0.44 + a

where "a" is the known absolute error in the 0.44 value.  We replace the remaining equations in a similar way.  Furthermore, since PA, PB, and PO are probabilities, they should be between 0 and 1.  Thus, your problem amounts to solving a set of 14 nonlinear inequalities in the three unknows PA, PB, PO.

I think this is a nonlinear optimization problem.  Unfortunately I don't know much about the subject to offer you some help here.  Perhaps others who know more can comment.

@Preben Alsholm 

a/x^b produces a pretty good fit:

q:=Statistics:-Fit(a/x^b,data,x);

@hhoang1 OK, then. Consider:

Plug in x=3.  What is the value of y?

Do it on paper, not Maple!

Does this throw some light on the problem?

@hhoang1 That depends on where you put the missing parenthesis.

1. The mathematical expression that you have shown has four opening parentheses and three closing parentheses.  Needs to be fixed.
2. You forgot to say over which range of x values you need to plot.