Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

How do I calculate and plot an Orthogonal Trajectory on Maple 2024?

Here's the equation of the contour lines:

x*y^2 - x^2 - y^2 = k

I need to make it pass through a precise point on my contour lines graph, and everything I do doesn't seem to work.

I have tried to understand why ODESteps cannot find a solution for cases where an ODE can separated into a polynomial expression and an ODE.

For example, when I trace like this

restart;
trace(Student:-ODEs:-ODESteps:-ModuleApply);
printlevel:=10;
Student:-ODEs:-ODESteps:-ModuleApply(x*(diff(y(x), x)) = 0);
printlevel:=1

trace is tracing an unknown procedure

and not showing any further details of this procedure.
What unknow function/procedure is traced?

Can, in this particular case, the message "Cannot compute integral"   be traced (it’s not an error message, which are traceable).

I have tried to increase printlevel which reveals some unkown procedures but did not get down to this one. Hence my questions.

sol := y = -3283/4253 - (3283*x)/4253, How can I determine the value of the coefficient of x?
How can I take the value of the coefficient of x? Thank you.

At here https://mathematica.stackexchange.com/questions/297104/how-can-i-convert-a-list-containing-three-points-and-its-equation-passing-three is discussing about "How can I convert a list containing three points and its equation passing three points to text file?" With mylist

mylist := [[[-12, 2, -1], [-11, 1, -5], [-10, -2, 3], 10*x + 6*y + z + 109 = 0], [[-12, 2, -1], [-11, 1, -5], [-10, 6, 3], 2*x - 2*y + z + 29 = 0], [[-12, 2, -1], [-11, 1, -5], [-9, 5, -7], 3*x - y + z + 39 = 0]]

How can I get the LaTeX file like this?

\documentclass[12pt,a4paper]{article}
\usepackage[letterpaper,margin=1.2in]{geometry}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label=\arabic*)]
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-10; -2; 3)$,\quad $(P) : 10 x+6 y+z+109=0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-10; 6; 3)$,\quad $(P) : 2 x-2 y+z+29=0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$,\quad $C(-9; 5; -7)$,\quad $(P) : 3 x-y+z+39=0$
\end{enumerate}
\end{document}


 

Given the ode   y''(x)*y'(x)=0, clearly it has solutions for y''=0 and y'=0.

These are y=c1+x*c2 and y=c1. But Maple gives 3 solutions

12592

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

ode:=diff(y(x),x$2)*diff(y(x),x)=0;

(diff(diff(y(x), x), x))*(diff(y(x), x)) = 0

dsolve(ode)

y(x) = c__1, y(x) = -c__1*x+c__2, y(x) = c__1*x+c__2

dsolve(diff(y(x),x$2)=0)

y(x) = c__1*x+c__2

dsolve(diff(y(x),x)=0)

y(x) = c__1

 

 

Download why_3_solutions_may_24_2024.mw

Where did the third solution come from? The 3 solutions are correct ofcourse, but why 3? There should only be two. 

FYI, I am using 

Physics:-Version()

The "Physics Updates" version in the MapleCloud is 1746. The 

   version installed in this computer is 1745 created 2024, May 

I have to check earlier Maple versions to see if same thing happens there.

I am stuck this command works seemlessly in Maple:

ThermophysicalData:-CoolProp:-Property(D, T = 20*Unit('degC'), P = 760*Unit('mmHg'), water)

but it does not work in Maple Flow. Does anyone knows why? Thank you so much for your help in the matter.

Why does 'simplify' not work when calculating Eigenvectors? Further, how can we express (2) in a more simplified form by using 'simplify'?simplify.mw

Is there a shortcut in Maple that will comment out a block of highlighted code? I tried searching on Mapleprimes but everytime I search I get a page generation error.

Hello :) 

I have a math problem, where I first need to use Linear regression to find the equation based on a set of data. I did that, no problem. 

However, in the next part of the problem I need to check if the residuals are under "normal distribution". Usually, I check if a dataset is normally distributed via "QQ-plot", and there will be no problems. But this time, because I need to check the residuals, I need to use the "residualQQplot(data,LinReg)" command to make it happen. But when I read the mean-value, mu, it says "-0," and nothing else? I know it should be "-3,2752*10^-15. 

The standard deviation is correct.

How do I fix this, so the residualQQplot shows me the right result? 

I have attached the worksheet here. worksheet_-_linear_reg_and_residuals_for_normal_distribution.mw

Thank you! 

what is the mathematics behind isolve? How can one selct special solutions?

Hello :) 

I don't know how to further simplify a complex equation as a solution to a differential equation. I have already tried "simplify(%)" etc. but it only repeats the same equation. I also know that it can be simplified to a much more "nice" equation in WordMat and TI-Nspire. I have attached a screenshot of my work. 

Thank you! 

with(Gym)

with(plots)

with(DEtools)

 

I need to find the solution to the differential equation: y'=0.0768*f(x)^2/3-0.0102*f(x) when f(1)=59. 

 

dsolve({diff(f(x), x) = 0.768e-1*f(x)^(2/3)-0.102e-1*f(x), f(1) = 59}, f(x))

f(x) = 2097152/4913+(384/4913)*(16384-4352*59^(1/3)+289*59^(2/3))*exp(17/2500-(17/2500)*x)+(1/4913)*(835584*59^(1/3)-110976*59^(2/3)-1807285)*exp(51/5000-(51/5000)*x)+(49152/4913)*(17*59^(1/3)-128)*exp(17/5000-(17/5000)*x)

(1)

simplify(%)

f(x) = 2097152/4913+(384/4913)*(16384-4352*59^(1/3)+289*59^(2/3))*exp(17/2500-(17/2500)*x)+(1/4913)*(835584*59^(1/3)-110976*59^(2/3)-1807285)*exp(51/5000-(51/5000)*x)+(49152/4913)*(17*59^(1/3)-128)*exp(17/5000-(17/5000)*x)

(2)

 

Download worksheet_.mw

I can't understand this behavior. Any idea why it happens?

Solve is able to solve equation   f(y)=x+A for y, but can't solve   f(y)=x for y.

This is unexpected for me. I do not see why it can solve it when RHS is x+A but not when RHS is just x.


 

21040

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1745. The version installed in this computer is 1744 created 2024, April 17, 19:33 hours Pacific Time, found in the directory C:\Users\Owner\maple\toolbox\2024\Physics Updates\lib\`

restart;

21040

sol:=int(1/sqrt(sin(y)),y);
solve(sol=x,y)

(sin(y)+1)^(1/2)*(-2*sin(y)+2)^(1/2)*(-sin(y))^(1/2)*EllipticF((sin(y)+1)^(1/2), (1/2)*2^(1/2))/(cos(y)*sin(y)^(1/2))

Warning, solutions may have been lost

sol:=int(1/sqrt(sin(y)),y);
solve(sol=x+b,y):
{%}; #to eliminate duplicates

(sin(y)+1)^(1/2)*(-2*sin(y)+2)^(1/2)*(-sin(y))^(1/2)*EllipticF((sin(y)+1)^(1/2), (1/2)*2^(1/2))/(cos(y)*sin(y)^(1/2))

{arctan(JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))^2-1, -(1/2)*JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))*(4-2*JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))^2)^(1/2)*2^(1/2)), arctan(JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))^2-1, (1/2)*JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))*(4-2*JacobiSN(((1/2)*I)*2^(1/2)*(x+b), (1/2)*2^(1/2))^2)^(1/2)*2^(1/2))}

 


I can trick it to solve  f(y)=x for y  by asking it to solve f(y)=x+A for y and then set A=0 in the solution. But one should not have to do this. Is this a bug or Am I missing something?

Download why_solve_when_adding_term_only_may_22_2024.mw

Dear Colleague. 

I am trying to improve the results of abs(res[i] - exy) in the following codes.

restart;
Digits := 30:

# Define the function
f := proc(n)
    -0.5*y[n] + 0.5*sin(x[n] - Pi)
end proc:

# Define equations
e1 := y[n+2] = 2*h*delta[n] + y[n] - h^2*(-2*sin(u)*f(n)*u^2 - 2*sin(u)*f(n+2)*u^2 + 2*sin(2*u)*f(n+1)*u^2 + 2*cos(u)*f(n)*u - 2*cos(u)*f(n+2)*u + 2*cos(2*u)*f(n+1)*u - 2*cos(2*u)*f(n)*u - 2*sin(u)*f(n) + 2*sin(u)*f(n+2) + sin(2*u)*f(n) - sin(2*u)*f(n+2) - 2*f(n+1)*u + 2*f(n+2)*u)/((2*sin(u) - sin(2*u))*u^2):
e2 := y[n+1] = h*delta[n] + y[n] - (1/2)*h^2*(-sin(u)*f(n)*u^2 - sin(u)*f(n+2)*u^2 + sin(2*u)*f(n+1)*u^2 + 2*cos(u)*f(n)*u - 2*cos(u)*f(n+2)*u + 2*cos(2*u)*f(n+1)*u - 2*cos(2*u)*f(n)*u + 4*sin(u)*f(n+1) - 4*sin(u)*f(n) - 2*sin(2*u)*f(n+1) + 2*sin(2*u)*f(n) - 2*f(n+1)*u + 2*f(n+2)*u)/((2*sin(u) - sin(2*u))*u^2):
e3 := h*delta[n+2] = h*delta[n] + h^2*(2*sin(u)*f(n)*u + 2*sin(u)*f(n+2)*u - 2*sin(2*u)*f(n+1)*u - 2*cos(2*u)*f(n+1) + cos(2*u)*f(n) + cos(2*u)*f(n+2) + 2*f(n+1) - f(n) - f(n+2))/(u*(2*sin(u) - sin(2*u))):

with(LinearAlgebra):
epsilon := 10^(-10):
inx := 0:
ind := 1:
iny := 0:
h := 0.01:
n := 0:
omega := 1:
u := omega * h:
tol := 1e-4:
N := solve(h * p = 8 * Pi, p):

err := Vector(round(N)):
exy_lst := Vector(round(N)):

c := 1:
for j from 0 to 2 do
    t[j] := inx + j * h:
end do:

vars := y[n+1], y[n+2], delta[n+2]:

step := [seq(eval(x, x = c * h), c = 1 .. N)]:
printf("%6s%15s%15s%16s%15s%15s%15s\n", "h", "Num.y", "Num.z", "Ex.y", "Ex.z", "Error y", "Error z");

st := time():
for k from 1 to N / 2 do
    par1 := x[0] = t[0], x[1] = t[1], x[2] = t[2]:
    par2 := y[n] = iny, delta[n] = ind:    
    
    res := eval(<vars>, fsolve(eval({e1, e2, e3}, [par1, par2]), {vars}));

    for i from 1 to 2 do
        exy := eval(sin(c * h)):
        exz := eval(cos(c * h)):
        printf("%6.5f%17.9f%15.9f%15.9f%15.9f%13.5g%15.5g\n", h * c, res[i], res[i+1], exy, exz, abs(res[i] - exy), abs(res[i+1] - exz));
        
        err[c] := abs(evalf(res[i] - exy));
        if Norm(err) <= tol then 
            h := 0.1 * h * (c + 1) * (tol/Norm(err))^(0.2);
        else 
            break
        end if;
        exy_lst[c] := exy;
        numerical_y1[c] := res[i];
        c := c + 1;
    end do;
    iny := res[2];
    ind := res[3];
    inx := t[2];
    for j from 0 to 2 do
        t[j] := inx + j * h;
    end do;
end do:
v := time() - st;
v / 4;
printf("Maximum error is %.13g\n", max(err));
NFE = evalf((N / 4 * 3) + 1);

# Get array of numerical and exact solutions for y1
numerical_array_y1 := [seq(numerical_y1[i], i = 1 .. N)]:
exact_array_y1 := [seq(exy_lst[i], i = 1 .. N)]:

# Get array of time steps
time_t := [seq(step[i], i = 1 .. N)]:

# Display graphs for y1
with(plots):
numerical_plot_y1 := plot(time_t, numerical_array_y1, style = point, symbol = asterisk, color = blue, symbolsize = 20, legend = ["TFIBF"]);
exact_plot_y1 := plot(time_t, exact_array_y1, style = point, symbol = box, color = red, symbolsize = 20, legend = ["EXACT"]);

display({numerical_plot_y1, exact_plot_y1});
Error_plot_y1 := plot(time_t, err, style = line, symbol = box, tickmarks = [piticks, decimalticks], color = navy, labels = [`h=Pi/8`, typeset(`Absolute Errors`)]);

I am suspecting that I didnt update the new h properly (I may be wrong, though). Please kindly help modify the code to allow the values of abs(res[i] - exy) to about 10^(-11). Thank you and best regards.

I am trying to solve the following recursion for any n, given a constant c. Here is my code for it:

c := 2:

A[i] := rsolve({a(0) = 1/n, a(i) = ((n - i + 1)/(n - i) + 1/(c*(n - i)))*a(i - 1)}, a);

total := evala(Simplify(sum(eval(A[i], i=k), k=0..n-1)));
evalf(eval(total, n = 6));

For c = 1, I get a valid (and correct) output, however for c = 2 for example, rsolve is returning A[i] = -GAMMA(-n)*GAMMA(-n + i - 1/2)/(GAMMA(-n - 1/2)*GAMMA(-n + i + 1)), which does not make sense when n is an integer. Is there something I am doing wrong here? Not sure why this is happening. Thanks!

Include print level in procedure
In the procedure code printlevel is not accepted
-enviroment variable
-interface variable 
error message : Error, (in interface) unknown interface variable, printlevel

More convenient in my opinion is to include on the printlevel depth in the procedure call?

restart;

fac := proc(n::integer)
    local previous_printlevel, result;
    previous_printlevel := interface('printlevel');  # Correct way to get the current printlevel
    interface('printlevel' = 3);  # Correct way to set the printlevel

    # De recursieve berekening
    if n = 0 then
        result := 1;  # Basisgeval
    else
        result := n * fac(n - 1);  # Recursieve aanroep
    end if;

    interface('printlevel' = previous_printlevel);  # Restore the original printlevel
    return result;
end proc;

fac(5);

proc (n::integer) local previous_printlevel, result; previous_printlevel := interface('printlevel'); interface('printlevel' = 3); if n = 0 then result := 1 else result := n*fac(n-1) end if; interface('printlevel' = previous_printlevel); return result end proc

 

Error, (in interface) unknown interface variable, printlevel

 
 

 

Download MP_vraag_printlevel_in_procedure_-lukt_niet.mw

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