It's hard to tell without further information.  It will help if you upload your worksheet here so that people may have a look.

In the screen where you edit your message, click on the big fat green arrow to upload your worksheet.

Linux 64 bit, Maple 2017.3 64 bit — crashes.

Mahmood, In the title you refer to "two equations" and later you ask about "these equations" but I can't tell what equations you are talking about.

To receive helpful answers, try to clearly state the problem that you are attempting to solve, and how you intend to solve it.  Then the rest of your calculations may make better sense.

@John Fredsted The M that enters the right-hand side of the differential equation is not the mass of Mercury, it is the mass of the sun, and therefore M = 1,99*10^30 is correct.

As to h, it is equal to r^2*diff(phi(t),t) which is a constant during the motion.  But r*diff(phi(t),t)  is the orbital velocity, let's call it v.  Therefore h = r*v.  We know r at perihelion, all we need is v at perihelion. A good approximation to v may be obtained by assuming that the orbit is circular, therefore the orbital velocity is the length of the orbit, that is 2*Pi*r divided by the time it takes to go around which is about 83 days if I remember it correctly.

@Seb1123 Your definition of f says that f(n) is the sum of all previous values of f.  If your suggested answer of 1,2,6,18,54 were correct, then 18=1+2+6, which may be true in some other universe, but not where I live.

If you really want to get the sequence 1,2,6,18,54, then you need to insert a factor of 2 in your formula, as in

f:=n->piecewise(n=0,1,n>=1,2*sum(f(k),k=0..n-1));

and then modify it as Joe Riel has explained.

@tomleslie Try a simple plot, such as
plot(x^2, x=-1..1);
and then export it as PDF.  You will see that the picture is placed in the upper left corner of a 11in x 8.5in paper, with a lot of whitespace around it.  I am absolutely at a loss to see the utility of that.  What in the world is the use of an  8.5'' x 11'' export?   Shouldn't exporting to PDF honor the plot's bounding box as exporting to all other image formats do?

I have complained to support@maplesoft about this repeatedly, and each time I have gotten an acknowledgement saying that, yes, that's dumb, and they will look into it, but nothing has happened over the last five or six years.

@tsunamiBTP For the integral of cos(n*t) Maple produces sin(n*t)/n, which is what is expected, as that is what we would normally do when we do the calculation by hand.  To be very pedantic, however, the proper answer should be

That's the way Macsyma did it in the 1980s, but that gets too tiring after a while. Maple only gives the general answer sin(n*t)/n. That works correctly most of the time, including in the case of the series calculation that I showed earlier.

But you are right in being super-careful about seeing that the software is doing what we expect it to do.

@Ramakrishnan That error message is due to badly formed expressions in your document. That is a common problem with Maple's Document mode and 2D input because these make such errors difficult to detect.  Many people prefer the Worksheet mode and 1D input for that reason.  You may want to give that a try. See:

Configuring Maple

Please note that this configuration will not convert any previous documents that you may have made under Maple's original settings. Sorry. It applies only to new worksheets that you will make after this configuration.

@Kitonum Parentheses and square brackets have different evaluation rules as you must certainly know; see

?Indexing Arrays, Matrices, and Vectors

That's the source of his problem.  He needs to change the parentheses to square brackets.

@carriewong Have a look at this modified worksheet.

Qmaple-modified.mw

@carriewong I think I know where your difficulty lies but it will be easier for me to explain the problem if you show me your equations first.  Then I will explain how to do it the right way.

The expression you have provided evaluates to a simple sum.  I see no brackets in the result.  You must be doing something else.

A note on Maple terminology:

An object of the form [a,b,c,...] (in square brackets) is called a list.
An object of the form {a,b,c,...} (in curly braces) is a set.

They behave quite differently.  Don't get them mixed up.

@tsunamiBTP 

Maple wouldn't yield the result = 1.862645149e-08 because the correct result is 1.931384967026*e-8 as I showed before.

As to why Maple produces 0. by default, consider this.  We have K = 25888158.4218766, therefore within 10 digits of accuracy, we have K^2 = 6.701967464e14.  Therefore within 10 digits of accuracy K^2 - 1 is the same as K^2 because taking away 1 affects digits beyond the level of accuracy that are being carried.  Consequently we get K = sqrt(K^2-1) = 0. which is the correct answer if we perform floating point calculations with 10 digits of accuracy.

If you increase the digits of accuracy, then you get the correct result of 1.931384967026*e-8, as I showed before.

Aside:

The problem that we are faced in that calculation is due to subtracting two large and almost equal numbers.  The usual way to avoid that is to note that

K - sqrt(K^2-1) = (K - sqrt(K^2-1)) * (K + sqrt(K^2-1)) / (K + sqrt(K^2-1))

which simplifies to 1 / (K + sqrt(K^2-1)).  This expression is algebraically equivalent to the original, however it now involves adding, rather than subtracting, two large numbers, and therefore it avoids the earlier numerical truncation issue.  With Maple's default 10 digits of accuracy we see that:

restart;
K := 25888158.4218766;
1 / (K + sqrt(K^2-1));
    1.931384967*10^(-8)

Many years ago George Pólya wrote a book titled How to Solve It. In it he says:

If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?

Which is the simplest finite difference problem that you know how to solve? Try that one first before you attempt this.

@vv By my note I meant to demonstrate a curious construction rather than a serious solution.  Your solution is definitely the right answer to the question.