## Ramakrishnan

Ramakrishnan Vaidyanathan

## 399 Reputation

9 years, 129 days

## Social Networks and Content at Maplesoft.com

With twenty years of Industrial experience and twenty years of teaching experience, I am now as retired Professor, using Maple to teach mathematics subject for students studying X to XII standards. Published XII Mathematics books.

## Location of axis label...

Just right click the graph anywhere,

A menu box will pop up.

Select axes (left click) Another nine line menu pops up.

Select labels (left click)Another four  line menu pops up.

Select edit vertical (left click)

Enter as many spaces as you want after the Ubar. You will get this Ubar distanced from y axis. In the same way enter the enter key as many times as you want the distance below x axis.LabelPosition_in_plottingVRK.mw

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## Removing items from favorite pallette...

Right click the item you want. A display will come with two options as follows

Remove from facorite palette

Left click the one desire.

If you want to gack to original click undo in the main  tool bar.

Hope it is useful.

## Three PDE solution...

Though i do not know much about the details of the problem, the doc gives an answer if we remove the ri and give ro only in the int fn.

I attach the doc. Hope it gives some insight into it. ro is a fn of ri which ia where there may be a difficulty. If ri and ro are independently specified, it may be easy for maple!.

I may be wrong. I only attempted to remove the error message.

Hope it is useful.

Ramakrishnan VThreePDEfor_site.mw

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 > partialequations:
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 > # BOUNDARY CONDITIONS
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## solving inequality...

I thought of obtaining the answers by eval(eq,[0.1,6]) and plot builder commands.

I get the answer as x = -0.4 to 0.1 and y >6.2 to 6 (close to y axis) are the range for eq >0. The rest are all negative field for eq. The conclusion is left to you from the graph.

I attach the doc for reference. where in i have also found the values of eq for sample coordinates.

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solveee_vrk.mw

 > restart:
 > with(SolveTools[Inequality]):
 > eq:=1/(x*y^(2/3))*8.620689655172415*10^(-16)*(-3.11*10^23*x^2*y^(7/6)-3.92*10^19*y^(25/6)+2.14545039999999*10^29*(0.0108*exp(-45.07/y)+exp(-19.98/y^(1/3)-0.00935317203476387*y^2)))/(x+0.015*y^(1.2));
 > eval(eq,[x=0.1]);
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 > eval((1)
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 > plots:-implicitplot(.8620689655e-15/x/y^(2/3)*(-.3110000000e24*x^2*y^(7/6)-.3920000000e20*y^(25/6)+.2317086432e28*exp(-45.07/y)+.2145450400e30*exp(-19.98/y^(1/3)-.935317203476387e-2*y^2))/(x+.15e-1*y^1.2),              x = -10.0 .. 10.0, y = -10.0 .. 10.0)
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## 4 equations and 4 variables - is it so d...

The problem i understand was you first check whether an answer is possible and then you start finding an answer.

Here concavity checks the possibility and equating the differentials give the answer.

I am now surprised that there is a method which gives the answer directly(!). It proves that maximum exists and hence concavity as well exists.

I checked the maple doc  for solving the four differentials with respect to the four variables and am yet to get the evaluation completed (half an hour passed). If what was found is correct by global optimisation package, then, maple can as well use this techniquefor solving atleast this particular case of four equations with four variables.

Attached maple struggling document in maple. (or may be it may take more than I can wait with patience for an answer)

We can not complain that maple has not solved it. It takes longer time and we lose patience and concavity_proof_question_(1).mw

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interrupt to cancel evaluation.

Cheers.

Ramakrishnan V

## First and second derivative tests...

There are two methods and the second one is comfortable in this case.

The methods are (in terms of conditions)

(1) If there exists a positive number r such that for every independent variable x in (a − ra) we have f′(x) ≥ 0, and for every x in (aa + r) we have f′(x) ≤ 0, then f has a local maximum at a.

(2) If f"<0,  then f has a local fmaximum at x (independent variable value)

Hence find second derivative of both TP1 and TP2 in terms of the four independent variables  T,E,W,p separately and if all of them are negative, then you have an optimum f value. Else only those variable that give f"<0 behave such as to give you an optimum f.

Cheers.

Ramakrishnan v

## Multiple plote...

Name the plots and use display plots command.

Also in the following command, sol1b_1 is not properly spelled.

gr1c:=multiple(plot[soln1a,soln1b1],dom1,color=[blue,purple])

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## Linear Fit...

Hope the following commands also would be ok.

Ramakrishnan V

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## Solve Algebraic Equations...

If I am correct in understanding that there are 12 equations with as many  variables, I just put the equations and variables, the Maple gives the answers. However few of them are interdependent and can be simplified (of course  we should solve once more the resultant variables in equation form).

Hope the answers satisfy you. Ramakrishnan vsolving12Equns.mw

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## Integration Problem...

Maple2016 does perfectly well this problem.

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Ramakrishnan V

rukmini_ramki@hotmail.com

## Hope this is acceptable. ...

Hope this is acceptable.

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 > plot(convert(F, list), x = 0 .. 10,  color = [red, black], scaling = constrained);

Ramakrishnan V

rukmini_ramki@hotmail.com

## Solution obtained...

The differential equation is

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with boundary conditions

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Isolating   and substituting x = 0, we get y'(0) interms of 'b' and thus 'a' is eliminated.

=

Result  now is

Substituting the value of  in terms of b and solving for y(x) we get an additional constant _C1. Applying bounadry condition y(0) = b, we get _C1 in terms of b.

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Substituting the value of _C1 we get the final solution as

Verification

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The solution is valid for any value of b including unit value. Substituting b = 1, the solution is

Correct Answer in my opinion is

I hope you are satisfied with my solution in attached document and reproduced below. Maple experts may obtain the same solution in easy commands.

Thanks for the problem.

Ramakrishnan V

rukmini_ramki@hotmail.com

## solution to trigonmetric equation...

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Aliter substitute the equation itself in the command as follows

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Ramakrishnan V

rukmini_ramki@hotmail.com

## (1) (2) ...

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=

Ramakrishnan V

rukmini_ramki@hotmail.com

Example 1

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Example 2