look at parfrac in help pages

@janhardo 

Linear or non-linear odes:  is there difference for trying to get parameteric solution for both types of ode?

For linear ode's, parametric solution is not needed, since almost all such ode's can be solved directly (either explicit or implicit).

Only time to resort to parametric form of solution method is for non-linear ode's. Lets conside only first order for now. Parametric solution will have the form   

          y(x)=f(t,...)
          x(t)=g(t...)

In the above, "t" is the parameter. So we have two equations, and one parameter that holds them.

If we can eliminate the parameter "t", say using the eliminate command in Maple or most likely by solving for t as function of x from the second equation using RootOf and replace that in first equation, then we get explicit solution.

But most of the times this is not possible to do, or it can make the solution much more complicated. So in this case, some books leave the ode solution as above in parametric form. 

Also sometimes solving the ode parametric form can actually be simpler than using direct approach for nonlinear odes'. 

But the bottom line,  parametric solution is a way to make it possible to express solution of  nonlinear ode's where other direct method are not possible or produce much more complicated answers. 

Here is one example from the handbook of exact solutions

ode:=y(x)*diff(y(x),x)-y(x)=-6/25*x+2/25*A*(2*sqrt(x)+19*A+6*A^2/sqrt(x));

Book solution is

The parameter above is tau.  You see, the solution is easy to read (one line!) , but this is because it is parametric. Most such odes can be hard or not possible to rewrite as explicit or even implicit form. But possible as parametric. 

As far as I know, Maple dsolve have no option to obtain/ask for parametric solution, in the sense the above book gives. 

Maple solution for the above is below. You see it is more complicated (specially the explicit one) but on the other hand, it is not parametric, so no need to try to solve for the parameter

Download parametric_sol.mw

The above book has many such ode's I found which Maple 2025 dsolve can't solve, but book has solutions for in parametric form.

@janhardo 

Not sure I understand the question. If you mean how to ask Maple for implicit solution to an ode, then the option 'implicit' is used for this.

And to prevent Maple dsolve from evaluating integrals internally while it is solving an ode, the option 'useInt' is used.

@Kitonum 

I understand all that about expand.

The issue is this is done in code. Not looking at screen.

So code has to keep trying different things and then pick the smallest (in leaf count) obtained. That is why I wrote the above full_simplify() command.

My point is that simplify() should just have worked here.  One does not have to use tricks or try 100 different combinations of commands to obtain simpler result.

Simplify is the most important core command in any CAS system. And Maple is supposed to be the most powerful symbolic CAS there is. Yet, I find it is always a struggle to obtain simpler result using just the simplify command.

Maple needs a new full_simplify() command, where it tries much much harder so users do not have to resort to tricks and different options hoping to get simpler result.

@Kitonum 

Ok, will add expand to the list I have to try. 

But any idea why simplify did not do it? Given that B is "simpler" in the sense it is smaller in size. (and also more clear to read).

@Scot Gould 

There seems to be something not setup right on your Maple? I just now did the command and worked fine

@vv 

But if there is no algorithm, how would you suggest the other software did it? (hard to trace what the other software does internally).

@janhardo 

Thanks, but on Maple 2025 I do not get what you show

It looks then this is regression in Maple and you must be using earlier version. Should then this be considered a regression bug?

@Scot Gould 

But, 50 years after the arrival of WYSIWYG, I strongly suspect Knuth doesn't believe we should still have to write hieroglyphics to obtain good mathematical symbols. 

Nothing will replace Latex/Tex for another 100  years. Try to submit a paper as a Maple document and see what happens.

Same for Mathematica notebooks. No one will accept these. These have to be converted to Latex.

People have been saying this about Latex for the last 30 years and it is now more in use than ever.

But Maple generates good Latex. So all what one needs to do is simply use that Latex, export it to file, and make Latex document of it at the end. Then compile it to PDF and HTML also if needed.

PDF is the universal document format for exchange between people. Not Maple and not notebooks  and not word documents or anything else. Anyone can open a PDF document, even browsers can read PDF document, but not everyone can open a Maple document as not everyone has Maple. 

One needs to know little Latex but not much. This is what I do. Everything I generate from Maple is Latex. Writing everything to file. I just have to add the surrounding text and all the the math env. to put the latex in (such as align or such). Maple does all the hard work in converting the equations to Latex. 

For manual writing, (For example,  making HW or school report or paper) there are two solutions to avoid writing in Latex and still at same time get Latex as output. There is a program called Lyx   

LyX combines the power and flexibility of TeX/LaTeX with the ease of use of a graphical interface.

More screen shots are at the above link. You can then export the document to Latex., compile it to PDF and also to HTML using tex4ht. Many people write their Phd thesis in Lyx. So it must be good. I tried it and it is good. I use SW 5.5 myself only because I am more used to it and been using it for long time.

Another, which is what I use, is scientific word 5.5. Unforuntually this program is no longer available for sale since the original US based company closed few years ago, but there is UK company that still sells it. You get 30 days trial. Make sure to use 5.5 and not 6.0 version. Here is link   Version 5.5 is much better. All GUI, then simply save the document and you have you latex file there.

SW 5.5 is what I use for all my math writing. I use SW 5.5 much more than Maple and Mathematica and any other program combined. Without it, I am lost :). It is the first program I install when I move to new PC. 

I can still write direct Latex if I want, but I agree, GUI is much easier and faster and more fun. There are hard core Latex users at tex stackexchange who say one should not even use a GUI that generates Latex and must write all the Latex by hand. I am not one of those.

It is also much much better than anything word or indesign can provide for math articles, or even without math, since nothing beats Latex for typesetting. I also think Tikz is very good, and you need Latex, but getting good at Tikz takes more time than even Latex.

Try making a 5,000 pages PDF book full of math using Word document or Maple. Good luck with that.

@salim-barzani 

when we solve by Dsolve option he just give us 1 solution and no more

I am not sure I understand what are you asking. dsolve picks a method to solve the ode, and returns the solution(s) found. Ofcourse an ode could be solved by many different methods (depending on the ode). But Maple only solves it using one method. Once it finds a method that can solve it, it does not solve it again using other methods.  If you set infolevel[dsolve]:=5; then you will see what methods it tries until it finds one.

Is this what you mean by one solution and no more? 

 i am looking for generate series solution but i need to watch what exactly they did and how find out 

Do you mean you want to see step by step how the series solution for an ode was found?

@Carl Love 

I am not sure I follow what you mean when you say "when the low-order terms are already an exact representation of the function". May be you mean if there is only one term in the solution, then big O are not shown. 

But ok, I take it then this is just a Maple own convention. 

Using sympy, it does give the big O notation in the solution for the same problem.

>python
Python 3.13.2 (main, Feb  5 2025, 08:05:21) [GCC 14.2.1 20250128] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import * 
... x = symbols("x") 
... y = Function("y") 
... 
>>> ode = Eq(Derivative(y(x), x)+ y(x), 1+ x) 
>>> ics = {y(0):0}
>>> dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)

And it gives

Eq(y(x), x + O(x**6))

Maple solution is ofcourse correct, but I was expecting to see big O at the end of each series solution as always is the case except here and so was surprised when I saw this and thought it was a bug.

@aroche 

I do not know why these recursion error showed up. But I am now running V 15 of support tools, and these no longer show. I can't reproduce these.

So only issue is the integration question I had. You could ignore these errors. I had no idea what caused them now. 

Here is updated worksheet.

Thanks.

Download int_strange_result_april_27_2025.mw

@Christopher2222 

oh I think SupportTools:-Version(latest) might work 

yes it does. This is what I do to get latest versions of SupportTools, like with Physics update

fyi, Maplesoft have published similar document in 2015  

HowMapleComparestoMathematica.pdf

I have not seen an update to the above since 2015.  There was some interesting discussions about the above document at understanding-differences-between-maple-and-mathematica-in-examples-picked-by-ma

But discussing Maple at Mathematica forum will be biased towards Mathematica, and similarly,  discussing Mathematica at a Maple forum will be biased towards Maple. 

wrong forum. Try https://tex.stackexchange.com/