@tomleslie Yes that's what I want but could you sent me the code of it? 

@tomleslie I ignore it because you gave me an useless answer and I have fixed it so why should I reply your answer which has nothing to do with my work? You guys always think there are lots of mathematical or logical errors in my code but I tell you there's nothing wrong with it , and you guys don't believe it.

If you still don't believe it please see this code. hw2_finished.mw This is exactly what I want in this question and I had fixed it myself and you guys helps nothing.So why not skip this question and see my another question? Why you always think you guys are smarter than me? I just don't understand it. I am not good at maple doesn't represent I am not good at math.This is the mistake you guys had made.I know what I want is right mathematically so why should I go to another way you guide me?

This code is just what you thought to be 100% wrong logically several days ago and I didn't change it at all, just added something to it and it works pretty good. How can you explain that?Do you still think I have mathematical and logical error?

@acer I won't do anything you called as spamming anymore so could you give me an answer? 

@acer 

Because no one answer it . And even if it's the same question I always add some new code into it so the original code is closer to the answer.

If you guys just stop blamming me and answer this question I will stop. I get the answer why should I continue to post it ?

@tomleslie 

Forget about this code which I have corrected it.

Could you look at this link?

http://www.mapleprimes.com/questions/207270-Get-A-Warning-When-I-Use-Dsolve-To-Get

It's the same question but different code and I got a warning .

I think it's only one step to the answer.

@Doug Meade Yes I tried your original responese and it doesn't work. Your new answer has worked partially because I can get a good picture of I1(t) but still get some trouble of I2(t). This is the new code hw2_finished.mw 

There is no mathmatical mistake of the expression I2(t). What I want is exactly an integral with limits eta(i).

Because I2(t)=int(sqrt(2*(H(t)+omega*cos(x))),x=-eta(t)..eta(t)) means for different t, the limits is different. Actually the eta(t) is root of p(t)=0. what I want is the area of a closed orbit like the third picture in this paper.

http://www.mapleprimes.com/view.aspx?sf=206645_question/20151103_083236.jpg

p and q is the solution of the ode

 t* = 71.3 which satisfy H(t*)=omega(t*)

as you can see in the picture when p=0, I denote the value of q = eta(t).

So please trust me there's no mathmatical mistake I know what I am doing, this thing is my graduation project called adiabatic invariants.

Could you help me with the code of I2(t)? It's only 1 step to the whole picture. 

@Carl Love Could you give me the code ? I don't know how to make it a piecewise function

@Carl Love This is the odeplot version of the same question   

The mistakes should be something about the functions I1(t) and I2(t).

example_hw2.mw

@Doug Meade Sorry to wasted your time, I uploaded a wroung file. I tried your code but not work, it's my fault.

Could you read the code again cause I have replaced it with the right code.

And there should be some questions after #.  

@Preben Alsholm 

Could you please see the following link?

http://www.mapleprimes.com/questions/206976-How-To-Put-if--Into-Dsolve

It's the same question but now I am sure that in this code there is no problem of mathematica, pure problem of maple.

@Preben Alsholm 

Sorry. I understand what you mean but you don't understand what I mean.

I know how to write it in a sheet of paper but don't know how to write it as a code which works on maple.

This is the code according to your solution but I don't know how to write it as code correctly so it doesn't work.

I am asking for the code ,not the methmatic. So if you just tell me how to do that without code, it doesn't help at all because the mathmatic process is not my problem, code is.

This is the code. hw2....mw

@Carl Love 

It works now and I got the answer.

Thanks very much

@Markiyan Hirnyk 

I1 and I2 are both not constant function as the t grows the value of these functions change.

@Carl Love 

I just tried and it doesn't work , there's another error 

Error, (in signum) too many levels of recursion

Actually I want to put the pictures of  [I1,t=0..100] and [I2,t=100..200] together in one picture but I don't know how to do this .

@Preben Alsholm 

If you get time could you write a code at the base of my code?

That's difficult to me , I am not familar with maple, always get errors.

@Preben Alsholm 

I think you didn't get my point.

I want to get the integral using the solutiong of the ODE which is q(t).

Actually I got the solution and by given t I know the value of q(t), so cos(q(t)) should be a number and sqrt(2(H+w*cos(q(t)))) should be a function of t.

I think I don't need to translate the variable.

Am I wrough? Because I am a beginner of maple so I don't know how to do that.

This is just my thought.

And what is the limits of q ? Could you explain it?