i have two euations including integration which has two unkwnon x1 and x2.
how can i get these equations solved, thanks for the help.

fsolve_problem.mw

For this integro-differential equation,

Equation:= int[y'(x)* (x^2)/[(x^2)-1],x)  =  (int[sqrt(y(x)])^(-2/3)

Maple is able to obtain an exact intrinsic solution

from which an exact solution can be extracted, namely,

ExtrinsicSolution:= y(x) = sqrt(3)*(-8*_C1*x^(8/3) + 12*x^2 - 3)^(3/4)

My question concerns how was this solution obtained.

Even more, specifically, 'odeadvisor' suggests converting the

equation in question to the form

ode:= y = G(x,diff(y(x),x));

However, I cannot reconcile how this can be applied to an equation which

contains two integrals. (Regretably, I am not able to directly, attach my

Maple worksheet directly on to this sheet). The situation is that after

applying 'dsolve' to the above 'Equation', Maple comes back with an

intrinsic solution which can was used to obtain the 'ExtrinsicSolution' in 

the above.  So it is the missing steps between applyingthe dsolve command

to Equation and the intrinsic solution which MS provides which, in turn, leads

to the 'Extrinsic Solution' above. I would greatly, appreciate if anyone can 

fill in the missing steps.

I was trying this ode with Maple

Do you agree this solution is not correct by Maple?

restart;

ode:=diff(y(t),t)+y(t)=Dirac(t);
ic:=y(0)=1;
sol:=dsolve([ode,ic],y(t),method='laplace');

It gives  y(t) = 2*exp(-t)

But from the discussion in the above link we see this is wrong solution. Maple also does not verify it:

odetest(sol,[ode,y(0)=1])

[-Dirac(t), -1]

Would this be considered a bug I should report or not? Note this result is only when using Laplace method. The default method gives better solution.

ode:=diff(y(t),t)+y(t)=Dirac(t);
ic:=y(0)=1;
sol:=dsolve([ode,ic],y(t));
odetest(sol,[ode,y(0)=1])

Maple 2023.2.1

How to convert barycentric coordinates to cartesian ? Thank you

I want to approximate a positive function that is decreasing in Gamma, say f(Gamma), that is very complicated yet very smooth. I need this in order to obtain a tractable and compact version of its derivative, which enters in the partial derivative of another (very simple) function.

Along the way, three related questions emerge: Derivatives_and_Approximations.mw

Thanks a lot!

Hi,

I am attempting to illustrate various solids in Maple. How can I do this with ? (Figure 3 on my worksheet). Thank you

S5SolidQ.mw

Is there a way to manipulate an equation so that it is in the form of (Expression of Primary Variables)*(Expression of Secondary Variables)
In the example below from Video 1: Fast Analytical Techniques for Electrical and Electronic Circuits (youtube.com), the primary variables are R1 and R2

PS. When I type ctrl-v to insert an image, I always get 2 copies.

I got the proportional symbol to work once, typing "proportional" + CRTL + Space.  Went for wlak came back and could not get it to work at all.

Does it actually work or am I imagining things?

Greetings All,

This is an application for control theory, specifially using Maple to solve control problems in the area of Interconnection and Damping Assignment Passivity Based Control (IDA-PBC).

- Assuming two variables (iL and Vo), there is a potential function that I am trying to solve for called "Ha".  I have two equations here, and I want to solve for Ha using the pdsolve() command:  

eq1 := diff(Ha(iL, Vo), iL) = rhs(result[1]);
eq2 := diff(Ha(iL, Vo), Vo) = rhs(result[2]);
pdsolve( {eq1, eq2  } );

Once I do this, Maple gives me an expression for Ha that has arbitrary functions in it (I understand where these are coming from).  So far, so good.

--> In order to get help solving for these arbitrary functions, I also want to tell Maple some constraints.  For example:

"the Hessian matrix of Ha must be positive definite"

Is there a way to do this?

with(geometry);
with(LinearAlgebra);
xA := 1;
yA := 0;
xB := 0;
yB := 0;
xC := 0;
yC := 1;
Mat := Matrix(3, 3, [xA, xB, xC, yA, yB, yC, 1, 1, 1]);
Miv := MatrixInverse(Mat);
phi := (x, y) -> Transpose(Multiphy(Miv), <x, y, 1>);
for i to 6 do
    B || i := phi(xA || i, yA || i);
end do;
Error, (in LinearAlgebra:-Transpose) invalid input: too many and/or wrong type of arguments passed to LinearAlgebra:-Transpose; first unused argument is Vector(3, {(1) = xA1, (2) = yA1, (3) = 1})
How to correct this error ? Thank you.

First, I should mention that I am new to Maple. When I try to solve the below system of ODEs

sys_ode := diff(x(t), t) = -x(t)^2/(4*Pi*y(t)*(x(t)^2 + y(t)^2)), diff(y(t), t) = y(t)^2/(4*Pi*x(t)*(x(t)^2 + y(t)^2))

with initial conditions of 

ics := x(0) = 1, y(0) = 1

using the command 

sol_analytic := dsolve([sys_ode, ics])

I receive the below error of 

Error, (in dsolve) numeric exception: division by zero

Any help or guidance to resolve this is greatly appreciated.

A ball on a turntable can move in circles instead of falling off the edge (provided the initial conditions are appropriate). The effect was demonstrated in a video and can be simulated with MapleSim. The amination below shows a simulation of a frictionless case (falling off the table) and the case with a coefficient of friction of one.

Also demonstrated in the video: Tilting the table leads to a sideward (not a downhill) movement of the ball.

The presenter of the video noted that in the untilted state, the ball eventually drifts off the table, attributing this to slippage. This drift is also observable in the animation above, where the ball starts moving in a spiral, whereas in a Maple simulation (below to the left), the ball follows a perfect circle. Only after optimizing contact and initial conditions, MapleSim produced a result (using the same parameters) that exhibits a similar circle, with a slight difference in angular orientation after completing two revolutions about the center of the circle.

Some observations on the MapleSim model:

• The results only slightly depend on the solvers. Numerical inaccuracies do not seem to be the reason for the difference in orientation. (Edit: see update below for the reason).
• The ball bounces up and down in the MapleSim simulation (0.0025 of the balls radius). The bouncing is caused by the fact that the initial position of the ball is above the elastic equilibrium position with respect to the table (the elastic contact makes the ball sink in a bit). Adding damping in the settings of the contact element attenuates this effect and reduces the drift.
• Drift is not observable anymore if in the contact element setting for "kmu" (smoothness coefficient of sliding friction) is set to larger values (above 10 in this example). This is an indication that sliding friction occurs during the simulation.
• Choosing the equilibrium position as initial condition for the ball does not prevent initial bouncing because MapleSim sets an initial velocity for the ball that is directed away from the table. I did not manage to enforce strictly zero velocity. Maybe someone can tell why that is and how to set MapleSim to start the simulation without vertical velocity. (Edit: see update below for the reason).
• Assuming an initial velocity towards the contact to cancel the initial vertical velocity set by MapleSim substantially reduces bouncing and increases the diameter of the circle. This finally leads to a diameter that matches the Maple simulation. Therefore the initial bouncing combined with slippage seems to dissipate energy which leads to smaller circles. Depending on the contact settings and initial conditions for vertical movement the diameter of the circle varied moderately by about 15%.

In summary, MapleSim can be parametrized to simulate an ideal case without slippage. From there it should be possible to tune contact parameters to closely reproduce experiments where parameters are often not well known in advance.  

Some thoughts for future enhancement of MapleSim:

• In the model presented here, a patterned ball would have been helpful to visualize the tumbling movement of the ball. Marking two opposite sides of the ball with colored smaller spheres is a fiddly exercise that does not look nice.
• A sensor component that calculates the energy of a moving rigid body would have helped identifying energy loss of the system. Ideally the component could have two ports for the rotational and translational energy components. I see professional interest in such calculations and not only educational value for toy experiments.
• A port for slippage on the contact elements would have helped understanding where slippage occurs. Where slippage is, there is wear and this is also of interest for industrial applications.

Turntable_Paradox.msim (contains parameter sets for the above observations)

Certainly a standard question.

I have an integer n*n matrix A (the entries are explicitly integers; there is no variable -type x- in the matrix). I want the Smith normal form of A, that is A=UDV where U,V are integer matrices with determinant +-1 and D is a diagonal matrix with -eventually- some zero and positive integers d_i s.t. d_i divides d_{i+1}.

"SmithForm()" doesn't work directly (I get rational -non integer- matrices). Maybe it is necessary to declare the matrix A as 'Matrix(integer)' ...
Thank you in advance for your help.

Hello,

In my the script a determinant of a matrix is set equal to 0 and then solved for. This formula has infinite solutions. So i will find the second root with: Student:-Calculus1:-Roots(Y, N = 0 .. limit, numeric) . This gives me several solutions and then i pick the second one as this is the critical buckling load of the column with the current parameters. 

Everytime i change a parameter i need to rerun a part of the script. I would like to generate a table where i define lets say parameter A and B and then run the script to give me value Ncr. then repeat this a 100 times with diferent parameters to see the influence of the parameters. 

Ive been looking around but i am not able to figure this one out. I think something like this should be possible? Thank you very much in advance!

Has anybody been able to get Maple working on macOS Sonoma 14.4. I know it's not supported yet, but I thought there might be someone who has solved the issue with the app crashing immediately after startup.