You would do this the same way you would enter it on a calculator (with a semi-colon added to the end):

(4*a^4-2*a^3+8*a-5) / (3*x^2-1);

But, there is nothing to do in this case.

If you meant to have the same variable in the numerator and denominator, then maybe you are interested in putting this quotient into "proper form" (with long division). If so, I do not know of a single command to do just this. (Maple developers: Hint, Hint! This would be a useful addition for students.)

Here are a couple of ideas about how you might get this information:

q := (4*a^4-2*a^3+8*a-5) / (3*a^2-1);
                               4      3          
                            4 a  - 2 a  + 8 a - 5
                            ---------------------
                                     2           
                                  3 a  - 1       
Q := quo(numer(q),denom(q),a):
R := rem(numer(q),denom(q),a):
q1 := Q + R/denom(q);
                                           41   22  
                                         - -- + -- a
                        4  2   2     4     9    3   
                        - a  - - a + - + -----------
                        3      3     9       2      
                                          3 a  - 1  
q2 := convert(q,parfrac,a);
                        4  2   2     4    -41 + 66 a 
                        - a  - - a + - + ------------
                        3      3     9     /   2    \
                                         9 \3 a  - 1/

It is pretty easy to see (in this case) that these two forms are equivalent.

Based on your question, I do not know your level of mathematical knowledge. The second approach, which finds the partial fraction decomposition, might be a topic you have not yet seen. Even if you have seen partial fractions, you need to realize that this approach will give a very different looking answer if the denominator factored (without square roots). For example:

q := (4*a^4-2*a^3+8*a-5) / (a^2-1);
                               4      3          
                            4 a  - 2 a  + 8 a - 5
                            ---------------------
                                    2            
                                   a  - 1        
Q := quo(numer(q),denom(q),a):
R := rem(numer(q),denom(q),a):
q1 := Q + R/denom(q);
                             2             -1 + 6 a
                          4 a  - 2 a + 4 + --------
                                             2     
                                            a  - 1 
convert( q, parfrac, a );
                      2                 5           7    
                   4 a  - 2 a + 4 + --------- + ---------
                                    2 (a - 1)   2 (a + 1)

I hope this has been helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

All of the previous suggestions are relevant, and appropriate, but they do not tell you exactly what you have to do. Clearly, part of your course involves you learning how to use Maple to solve problems. This reason prior respondents have been reluctant to give you a complete solution to your problem. (If I am wrong about this, I'm sure I'll hear about it.)

I am not going to give you any specific Maple help either. But, I will try to be a little more explicit about the work that you need to be doing.

How did you arrive at your answers on paper?

I will assume that you looked at the expressions and asked where the square roots will be real and, later, when determining distinctness, not zero. How did you do this? You need to ask the same questions to Maple.

The only Maple hints I will give are

1. To get the expression under the square root you can use the mouse to select the expression you want, or use the following command:

   expr := sqrt( 2-x^2 );
                                           (1/2)
                                   /     2\     
                                   \2 - x /     
   q := op(1,expr);
                                           2
                                  q = 2 - x 
   

   But, in your case it might be easiest to simply enter this expression explicitly. You can then ask Maple for help solving q>=0 or q>0 (see the help for solve).

    

2. To understand the output Maple is likely to give when you do the above, see the online help for RealRange (?RealRange).

Have you asked your professor, or anyone else associated with your course (even other students - if permitted), for any help with this?

Doug

All of the previous suggestions are relevant, and appropriate, but they do not tell you exactly what you have to do. Clearly, part of your course involves you learning how to use Maple to solve problems. This reason prior respondents have been reluctant to give you a complete solution to your problem. (If I am wrong about this, I'm sure I'll hear about it.)

I am not going to give you any specific Maple help either. But, I will try to be a little more explicit about the work that you need to be doing.

How did you arrive at your answers on paper?

I will assume that you looked at the expressions and asked where the square roots will be real and, later, when determining distinctness, not zero. How did you do this? You need to ask the same questions to Maple.

The only Maple hints I will give are

1. To get the expression under the square root you can use the mouse to select the expression you want, or use the following command:

   expr := sqrt( 2-x^2 );
                                           (1/2)
                                   /     2\     
                                   \2 - x /     
   q := op(1,expr);
                                           2
                                  q = 2 - x 
   

   But, in your case it might be easiest to simply enter this expression explicitly. You can then ask Maple for help solving q>=0 or q>0 (see the help for solve).

    

2. To understand the output Maple is likely to give when you do the above, see the online help for RealRange (?RealRange).

Have you asked your professor, or anyone else associated with your course (even other students - if permitted), for any help with this?

Doug

Be careful.

In your previous post you wrote:

Digits:=n     #will the number or precise digits to n

evalf(sqrt(x))  #will evaluate the floating point answer of sqrt(x) to a precision of n floating point digits.

Digits is the number of significant digits used in floating-point calculations, not the precision of the result. Depending on the situation, the actual number of digits that are correct can be (much!) less than the number of digits used in the calculation.

This is not an issue in many situations, but be aware that loss of precision can occur and take measures to develop confidence in the digits you do report.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

Be careful.

In your previous post you wrote:

Digits:=n     #will the number or precise digits to n

evalf(sqrt(x))  #will evaluate the floating point answer of sqrt(x) to a precision of n floating point digits.

Digits is the number of significant digits used in floating-point calculations, not the precision of the result. Depending on the situation, the actual number of digits that are correct can be (much!) less than the number of digits used in the calculation.

This is not an issue in many situations, but be aware that loss of precision can occur and take measures to develop confidence in the digits you do report.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

This approach has trouble because the defining equation (x/t) is not defined for t=0. Humans, of course, can overlook this difficulty, but Maple is not human.

I would start this problem by entering the differential equation and its direction field:

with( plots, DEtools ):
ODE := t*diff(x(t),t)-x(t)=0;
P1 := DEplot( ODE, x(t), t=-10..10, x=-5..5 ):

The isoclines are the curves where the derivative is a constant. Here, these curves are given by:

ISO := eval( ODE, diff(x(t),t)=C );

If this equation is simple enough, this equation could be solved for x(t). For example:

ISO2 := solve( ISO, x(t) );

A sample of these curves can now be plotted:

P2 := plot([seq](ISO2,C=[$1..3]/4),t=-10..10, color=green ):
display( P1, P2 );

If it is not possible to solve explicitly for x(t) in the isocline, implicitplot can be used, as follows:

ISO3 := eval( ISO, x(t)=x );
P3 := implicitplot( [seq](ISO3,C=[$1..3]/4), t=-10..10, x=-10..10, color=blue ):
display( P1, P3 );

These approaches should handle the majority of cases that you are likely to encounter.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

This approach has trouble because the defining equation (x/t) is not defined for t=0. Humans, of course, can overlook this difficulty, but Maple is not human.

I would start this problem by entering the differential equation and its direction field:

with( plots, DEtools ):
ODE := t*diff(x(t),t)-x(t)=0;
P1 := DEplot( ODE, x(t), t=-10..10, x=-5..5 ):

The isoclines are the curves where the derivative is a constant. Here, these curves are given by:

ISO := eval( ODE, diff(x(t),t)=C );

If this equation is simple enough, this equation could be solved for x(t). For example:

ISO2 := solve( ISO, x(t) );

A sample of these curves can now be plotted:

P2 := plot([seq](ISO2,C=[$1..3]/4),t=-10..10, color=green ):
display( P1, P2 );

If it is not possible to solve explicitly for x(t) in the isocline, implicitplot can be used, as follows:

ISO3 := eval( ISO, x(t)=x );
P3 := implicitplot( [seq](ISO3,C=[$1..3]/4), t=-10..10, x=-10..10, color=blue ):
display( P1, P3 );

These approaches should handle the majority of cases that you are likely to encounter.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

I agree that it is not always appropriate to try to use the same approach to a problem in different software systems. Each system has its own strengths and these need to be taken into account. The different approaches suggested for this problem are all reasonable.

But, nobody has yet presented what is the most appropriate way to approach this problem.

s := [$1..5000]:
plots[pointplot3d]([seq([x,sin(x),cos(x)],x=s)]);

Note that Maple makes a distinction between a set and a list. In most cases like this, a list (with order) is what users want. Sets (without order) do have their uses, but you have to be careful to select the right data structure for the right problem.

To close, I would note that I would create this specific plot without explicitly constructing the data:

plots[pointplot3d]([seq([x,sin(x),cos(x)],x=[$1..5000]);

I hope this is helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

I agree that it is not always appropriate to try to use the same approach to a problem in different software systems. Each system has its own strengths and these need to be taken into account. The different approaches suggested for this problem are all reasonable.

But, nobody has yet presented what is the most appropriate way to approach this problem.

s := [$1..5000]:
plots[pointplot3d]([seq([x,sin(x),cos(x)],x=s)]);

Note that Maple makes a distinction between a set and a list. In most cases like this, a list (with order) is what users want. Sets (without order) do have their uses, but you have to be careful to select the right data structure for the right problem.

To close, I would note that I would create this specific plot without explicitly constructing the data:

plots[pointplot3d]([seq([x,sin(x),cos(x)],x=[$1..5000]);

I hope this is helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

The classical approach to solving delay differential equations with delay a is to solve the equation on the interval [0,a] (this requires initial data on the interval [-a,0]). Then, this solution can be used to solve the problem on the next interval of length a: [a,2a].

You do not indicate the initial data for your problem, so I cannot illustrate this for your problem. This general approach does not assume the DEs are homogeneous or linear. In terms of Maple, you use only dsolve. The key is to realize that all of the information in the delay term is known provided you work on intervals of length a.

Provide explicit details about your initial data and I will try to show how I would implement this.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

The classical approach to solving delay differential equations with delay a is to solve the equation on the interval [0,a] (this requires initial data on the interval [-a,0]). Then, this solution can be used to solve the problem on the next interval of length a: [a,2a].

You do not indicate the initial data for your problem, so I cannot illustrate this for your problem. This general approach does not assume the DEs are homogeneous or linear. In terms of Maple, you use only dsolve. The key is to realize that all of the information in the delay term is known provided you work on intervals of length a.

Provide explicit details about your initial data and I will try to show how I would implement this.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

If you are interested in safety, why not think about the backup chute. It is smaller, and so will have a smaller k, but still larger than the initial free-fall resistance. You can figure out the latest time (lowest altitude) that the backup can be deployed to bring your landing velocity below some threshold. But, when do you have to attempt to open the primary chute so that you have time to deploy the backup chute in the case it is needed. As I recall, the general guidelines in the Air Force manual (reference in my papers), which are stated in terms of times after leaving the plane (at always the same altitude), match up very nicely with this. I cannot believe this is coincidence. Take a look for yourself.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

If you are interested in safety, why not think about the backup chute. It is smaller, and so will have a smaller k, but still larger than the initial free-fall resistance. You can figure out the latest time (lowest altitude) that the backup can be deployed to bring your landing velocity below some threshold. But, when do you have to attempt to open the primary chute so that you have time to deploy the backup chute in the case it is needed. As I recall, the general guidelines in the Air Force manual (reference in my papers), which are stated in terms of times after leaving the plane (at always the same altitude), match up very nicely with this. I cannot believe this is coincidence. Take a look for yourself.

Doug

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

Acer is correct. When I wrote 6G I was saying the force jumped by 6 times the force of gravity. That's more than 1000 lbs for an average adult male (don't forget the equipment contributes to the overall mass of the skydiver). When I did this work I found some textbook examples with truly unsurvivable situations (instantaneous forces in excess of 10G).

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

Acer is correct. When I wrote 6G I was saying the force jumped by 6 times the force of gravity. That's more than 1000 lbs for an average adult male (don't forget the equipment contributes to the overall mass of the skydiver). When I did this work I found some textbook examples with truly unsurvivable situations (instantaneous forces in excess of 10G).

---------------------------------------------------------------------
Douglas B. Meade
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/