If b, c, and x are positive, then how can x^2+b x + c be zero?

@aryaneh The solution is simpler than what you have shown.  See the attached worksheet for details.

doit.mw

I know nothing about fractional differential equation but you may find the following useful:

    http://www.sciencedirect.com/science/article/pii/S0377042712000064

Example 12 there is very much like your system of equations.

@fereydoon_shekofte Your formula is correct.  It agrees with what I had written in the paragraph under the heading Conclusion in the worksheet daylight3.mw which I had posted earlier. In your formula tan(tilt) appears in the denominator while in mine it appears in the numerator because you measure the tilt angle relative to the equator while I measure it relative to the north pole.

@vv Thanks for the information about Explore and your modified code.  I rarely think of using Explore but I see that in this case it can be quite helpful, especially when memory is tight.

As to your suggested parameter ranges to Explore, I would change lambda=0..2*Pi to lambda=-Pi/2...Pi/2 because that's how latitude is customarily measured.  Also a practical range to the tilt angle would be tau=0..Pi/2.  But these are only minor considerations.

@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?