Your sum does not output what you display below the code.   The code below does, without your coefficients in front of the c[0]'s.

>restart:

>y1 := sum(c[0]*x^(1/2+i), i = 0 .. 4);

>tmp1:=simplify(y1/(c[0]*x^(1/2)));

>ans:=c[0]*x^(1/2)*tmp1;

Check to see if it looks like the original:

>expand(%);
 

My "Default format for new worksheets:" under the Interface Tab is set to Worksheet.  I am not to clear on all of this, because I rarely use the Gui interface.  I hope I have not added confusion to the matter.  I think you have answered the origianl question though with your suggestions.  Thank you. 

Following the examples that Acer mentions above, I tried:

To change your default entry mode, select Tools->Options from the menubar, then select the Display tab in the pop-up that appears, and then use the "Input display" dropdown menu in that tab to choose "Maple Notation".

For some reason, this does not work for me.  After making the selection as stated above, I click the "Apply to Session" button.  I also tried it with the "Apply Gobally" button.  I shut down Maple and restarted.  I checked the settings and the display tab has the Input selection set to "Maple Notation".  The end result is the same, as the original poster stated when I enter the commands and solve with the space after the solve command.  I tried this before I replied to the question.  Can someone verify this for me.  Maybe I am missing something.

This is a difference between the  Classic Gui and the non Classic version.  In the Classic Gui version, you get the expected result.  This has to do with how the non Classic Gui handles spaces using 2D math.  Ideally, as you state above, there should be consistency between the two.  If it works in one, it should work in the other.  I am referring to Maple 12 here.  I have not tested this with other versions.  Either way,  I never insert spaces after a Maple command, regardless of interface type.

Take a look at the help files for ode's with initial conditions.  Type the following and hit the enter key, or use the help browser and look up dsolve.

>?dsolve/ICs

The line below will set up the problem for dsolve to solve directly.  Give it a try using your initial condition.

>ode1 := diff(C(t),t)=C(t)*(1-0.0002*C(t));

The answer using C(0)=2 should come out as:

C(t) = 5000/(1+2499*exp(-t))

Below is a template with data.  You can fill in your own.   I did it for 4 months.  You can expand it to include the rest.  January=1, Februray=2, March=3, April=4...

>restart:

>animal[frog]:=[[1,30],[2,20],[3,25],[4,27]];

>animal[snake]:=[[1,10],[2,15],[3,11],[4,19]];

> animal[spider]:=[[1,15],[2,18],[3,22],[4,24]];

> animal[beetle]:=[[1,29],[2,32],[3,35],[4,19]];

> with(plots):

> a:=listplot(animal[frog],color=blue,legend="frog"):
> b:=listplot(animal[snake],color=green,legend="snake"):
> c:=listplot(animal[spider],color=red,legend="spider"):
> d:=listplot(animal[beetle],color=yellow,legend="beetle"):
> display({a,b,c,d});
 

Regards,
Georgios Kokovidis

Break up your code into segments and see if you can get each one to execute indepedantly, before combing into one  long statement like "animate( plot,  [   [seq( traj[ k ],  k=1..J )]   ]  ,  J=[ 2,3,4,5] )".

For your particular example, because the data set is small, it is easier to do it manually.

> restart:with(plots):
> traj:= [[2,1], [3,2],[4,3], [6,8],[3,-2]];
> traj1:=([[2,1],[3,2]]);
> traj2:=([[3,2],[4,3]]);
> traj3:=([[4,3],[6,8]]);
> traj4:=([[6,8],[3,-2]]);
> a:=plot(traj1):
> b:=plot(traj2):
> c:=plot(traj3):
> d:=plot(traj4):
> display(a,b,c,d,insequence=true);
 

Right click on the plot, go to Animation, and them click on Play.

See attached worksheet.  Your approach of graphing is correct.  It gives insights into the proper approach for solving the problem.  Hope this helps.  

Download 221_value_of_k.mws
View file details

What kind of numbers?  There exists an infinite number of reals between your two numbers.  Are you looking for something else?

What kind of numbers?  There exists an infinite number of reals between your two numbers.  Are you looking for something else?

As G A Edgar mentions, there might not be a closed form solution for your problem. 

First question that I have is, "How did you derive this equation.?"  Can you show the original circuit problem that you solved to get you here.  What are the initial conditions at t=0?  The attached worksheet makes some assumptions for your values and uses the "numerical" option to solve your problem.  See the help pages for dsolve/numeric for more examples.

Download 221_diff_eq.mws
View file details

Take a look at the help files for the following commands:

Re - return the Real part of a complex-valued expression
Im - return the Imaginary part of a complex-valued expression
 

>?Re

will bring up the help page for both of these, with examples of usage.

Take a look at the following worksheet.  The last command will bring up the help file for the CurveFit package, which has more options.  The worksheet was created in version 9.52.  I tried it in Version 12 as well.  I think the CurveFit package does not exist in version 9.52.  If you have version 12 you can give it a try.  This is just to highlight a few examples that you can modify for your own needs.  It is by no means complete, or the best way to approach this type of problem.  I just wanted to illustrate that it can be done.

Download 221_curves.mws
View file details

In the future, if you have a large data file, load it into a worksheet and post the worksheet.  Your data was in the form of an image which makes it difficult to cut and paste it.

Substituting limits after the integration seems to work just fine, but that does not explain why it can't be done in one step.

>restart:

>ans:=int(sqrt(t^5+6*t)*(5*t^4+6),t);

ans := 2/3*t*(t^4+6)*(t^5+6*t)^(1/2)

>eval(subs(t=1,ans)-subs(t=0,ans));

14/3*7^(1/2)

 That is the right command for suppressing the warning.  I tried it and it works, using one of the examples in the help file.

I started my worksheet like this.

>restart:

> _Env_dsolve_nowarnstop := true:

> sys:={diff(x(t),t,t)=0,diff(y(t),t,t)+9.8=0}:
> IC1 := {x(0)=0,y(0)=10,D(x)(0)=10,D(y)(0)=40}:
> stopcond:= [y(t)-8*sin(x(t)),x(t)-100]:
> ans:=dsolve(sys union IC1, numeric, stop_cond=stopcond):
> ans(100);
 

Without the environment variable set to true, you will get the following warning:

"Warning, cannot evaluate the solution further right of 8.2569790, stop condition #1 violated"