I see no reason for such an error message in the code that you have shown.  When I paste that code to a Maple worksheet, it executes without a flaw.  Perhaps there is something else in your worksheet that is interfering.  Consider uploading your troublesome worksheet for comments.  Use the large green up-arrow in the reply screen to upload

Let

f := (x,t) ->  x + sin(x) +  ln(t);

Then consider f(x,t) = 0 as defining x(t) and a function of t.  Note that x(1)=0.

We have f(x(t),t) = 0,  therefore the derivative of f(x(t),t) with respect to t is zero.  This gives us a differential equation, as you have noted:

diff(f(x(t),t),t);
solve(%, diff(x(t),t));
de := diff(x(t),t) = %;

Solve the differential equation

dsol := dsolve([de, x(1)=0], x(t), numeric);

Now define

F := x -> x + x^2;

and plot F(x(t)):

plots:-odeplot(dsol, [t, F(x(t))], t=0..10);

Here is the solution to Case (vi) in the picture that you have posted.

The characteristic lines for Case (vi) with a=0, b=1, c=2

Another sample:

The characteristic lines for Case (vi) with a=-1, b=0.5, c=2

Your code is correct other than

h=evalf((xn-x0)/n);

should be

h:=evalf((xn-x0)/n);

Aside: To verify the correctnest of your integrator, try it on a cubic polynomial.  Simpson 3/8 should produce the exact answer. (Yours does.)  Remove the evalf()s to see that clearly.  Thus:

expand(simp38(x -> x^3, a, b, 9));

results in 

b^4/4 - a^4/4

as expected.

If you open the file in Maple, you will get everything that the user has typed.  You don't need a special command for that.  But perhaps I have misunderstood your question.

Your complexplot3d() is not producing a plot because complexpplot3d() is defined in the plots package, but you are loading the plots package too late.  Move the with(plots) line further up in the worksheet.

The comlexplot() at the worksheet's end does produce a graph.  To help you see the graph, change the plotting command to

complexplot({a, b}, numpoints = 100, color = red, thickness = 3, filled = true, title = "Stability Region");

What you have written in your message focuses on Maple.  That's not quite right.  I suggest that you set Maple aside for now, and make an effort to understand what Euler's method is supposed to do.  Remember: Euler lived about 250 years ago and he did not have Maple.

Only after you have understood what the method does, attempt to ask Maple to do that thing for you.

The answer is approximately 1/2015!.  Here is a sketch.

Let's write p(x) for mul(x+j, j = 0 .. 2015).  We are looking for the positive root of p(x)=1.  We have:

p(x) = (x^2015 + ... + 2015!) . x

We see that p'(0) = 2015!, thererefore the graph of p rises quite steeply(!) from p(0)=0 to p(t)=1, where t is the root you are looking for.  Approximating the graph by a straight line, we get t = 1/2015!  We note that

> 1.0/2015!;
                                    -5786
                      8.667800428 10     

which agrees withKitonum's alternative computation.

I don't see why you would want to modify the standard Laplace transform code since you can reduce your integrals to the usual Laplace transform through a change of variables.

The integral int(f(u*t)*exp(-s*t), t=0..infinity) after the change of variables u*t=tau takes the form

(1/u)*int(f(tau)*exp(-s/u*tau), tau=0..infinity),

which is the same thing as

(1/u)*laplace(f(tau), tau, s/u).

Similarly, the integral int(f(u*t)*exp(-t), t=0..infinity) changes over to

(1/u)*laplace(f(tau),tau,1/u).

Whether you want to to do these changes of variables or not depend on your ultimate goal.  In practice, both of those integrals may be computed without resort to any changes of variables:

int(f(u*t)*exp(-s*t), t=0..infinity) = laplace(f(u*t), t, s);

int(f(u*t)*exp(-t), t=0..infinity) = laplace(f(u*t), t, 1);

I don't quite understand what it is that you want to do.  The graphs you have shown are those of y versus psi.  You don't need the velocity V or the DEtools package to produce those.  You will obtain the first of your three diagrams if you replace your plot() command with

Change R=2.0 to 5.0 or 6.5 to obtain the other two diagrams.  By the way, the three diagrams that you have shown are identical.   They would be different  if they were done correctly.

Introduce velocities as new variables and convert your second order differential equations to first order.  Then you will have a system of four first order differential equations to solve, and events on velocities may be specified in a straightforward way.  See the attached worksheet.  Here is what we get:

Worksheet here: mw.mw

This plots p versus w:

As acer has noted, this may be a bug.  Or perhaps this is due to a not-so-well documented requirement in Eigenvectors().  I keep a copy of the old Maple 11 for testing purposes.  Running your problem in Maple 11 gives an error message which provides a clue:

> LinearAlgebra:-Eigenvectors(A);

Error, (in LinearAlgebra:-LA_Main:-Eigenvectors) expecting either Matrices of rationals, rational functions, radical functions, algebraic numbers, or algebraic functions, or Matrices of complex(numeric) values

"Expecting rationals"?  So, we change your 0.991 and 0.997 to 991/1000 and 997/1000, and voila, we get an answer! This works  equally well on Maple 11 and Maple 18.

It would be good if you showed at least some attempt toward solving your own homework problem.  Write down the equations.  Write down the initial and boundary conditions.  These have nothing to do with Maple, so "im totally new to maple" is not a good excuse.  Then try to code and solve your equations in Maple.  If you run into problems with Maple, then say where you are stuck and you will receive help.  But it's not fair to ask someone (not "some1") else to do your homework problem from scratch.

zohre your equations are badly scaled.  You have, instance, 

Have in mind that sinh() and cosh() are more or less like the exponential function.  Does raising e to the power of 100 make any sense in regard to the problem that you wish to analyze?  Probably not.

You need to rescale your variables so that they take values within a resonably "normal" range.  You wouldn't express the distance between Lodon and Paris in micrometers.  You wouldn't express the size of your computer screen in kilometers.  For the same reason, you shouldn't express your x(t) and y(t) in units that lead to strange coefficients that you have shown.

Try, as an exercise, the following rescaled version of your equations:

eq1 := diff(x(t),t) = sinh(x(t)+y(t)) - x(t)*cosh(x(t)+y(t));
eq2 := diff(y(t),t) = sinh(-x(t)+y(t)) - y(t)*cosh(-x(t)+y(t));
dsol := dsolve({eq1, eq2, x(0)=0, y(0)=1}, {x(t), y(t)}, numeric, range=0..10);