## Solutions may have been lost warning...

Hello everyone,

I am trying to solve 6 equations with 6 unknowns in maple. I tried to simplify them and they are shown below. Unfortunately, after using solve I get:

Warning, solutions may have been lost
sol := ()

Can you help me with it, please? I tried to solve it with Matlab and got a warning that no explicit solution could be found and had no luck with mathematica either.  Thank you very much!!

restart;

omega_1 := 1;
beta_1 = 0.2;
omega_2 := 0.15;
beta_2 := 0.15;
mu := 0.4;
g := 9.81;
K := 5;
vb := 0.5;
tc := arccos(vb/(A*omega));

Ns := mu*(omega_2^2 - K)*B*(2*omega*tc - sin(2*omega*tc) - pi)/pi;

Nc := 4*mu*(omega_2^2 - K)*x2*sin(omega*tc)/pi + mu*(omega_2^2 - K)*C*(2*omega*tc + 2*sin(2*omega*tc) - pi)/pi;

a0 := -mu*(omega_2^2 - K)*x2*(1 - 2*omega*tc/pi) + 2*mu*(omega_2^2 - K)*C*sin(omega*tc)/pi;

eq1 := -A*omega^2 + A*omega_1^2 - B*K - Ns;

eq2 := 2*A*beta_1*omega*omega_1 - C*K - Nc;

eq3 := -2*C*beta_2*omega*omega_2 - B*omega^2 + B*omega_2^2 - A*K;

eq4 := 2*B*beta_2*omega*omega_2 - C*omega^2 + C*omega_2^2;

eq5 := omega_1^2*x1 - K*x2 - a0;

eq6 := omega_2^2*x2 - K*x1 - g;

sol := solve({eq1, eq2, eq3, eq4, eq5, eq6}, {A, B, C, omega, x1, x2});

## How can Maple animate this family of pursuits?...

This worksheet contains an unnamed theorem on page 202 of David Wells's book The Penguin Dictionary of Curious and Interesting Geometry.

Somehow I have uploaded both its contents and (a) link(s) to it.

What Maple code can animate and display, in turn, each of the portrayed pursuit paths?

Pursuit_problem.mw

Consider a target point T which moves at constant speed along a straight line, and a moving point P which at all times moves directly towards T. If P starts anywhere on the outermost ellipse, and T starts from a focus of the outer ellipse, then P always captures T at the same point, the centre of the ellipse.

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The concentric ellipses, whose shape depends on the relative velocities of T and P, are isochrones, and the curves of pursuit are their isoclinal trajectories

## Help with code for impulsive optimal control probl...

Hi,

I am using Maple 2020 to numerically solve/generate numerical plots for my impulsive control problem.

The optimal control problem is:

T is time from0 to T where T is the terminal time

K(t), B(t) and M(t) are state variables

w(t) is a control variable

a(ti) is the impulsive control variable at time ti,

a(ti) \in [0,1] for i=1,2,…,N

ggamma, ttheta, ddelta1, ddelta 2, c1 and c2 are constants

K(T)=M(T)=0 B(T)>0

K'(t)=ggamma*K(t)*w(t)-ttheta*M(t)

B’(t)=ggamma*K(t)+ttheta*M(t)*B(t)

M’(t)=M(t)-ggamma*w(t)

M(ti)=M(ti-)+a(ti)M(ti)*ddelta1

K(ti)=K(ti-)-a(ti)K(ti)*ddelta2

Objective: maximize B(T)-integral from 0 to T of c1*(w(t))^2dt-sum i=1 to N of c2*a(ti)

Here is the code I enter to MAple:

restart;

# Define the constants
ggamma := 1.0;
ttheta := 2.0;
ddelta1 := 0.1;
ddelta2 := 0.2;
c1 := 0.5;
c2 := 0.3;
T := 5.0; # Terminal time

# Define the impulsive changes in M(t)
impulse_changes := proc (t)
local ti_values, imp_values, result;
ti_values := [1.0, 2.0, 3.0]; # Example impulsive time instants
imp_values := [0.2, 0.1, 0.3]; # Corresponding impulsive control values
result := 0;
for i from 1 to nops(ti_values) do
if t = ti_values[i] then
result := result + imp_values[i]*M(ti_values[i])*(ddelta1 - ddelta2);
end if;
end do;
return result;
end proc;

# Define the system of differential equations
diffeqs := {diff(K(t), t) = ggamma*K(t)*w(t) - ttheta*M(t),
diff(B(t), t) = ggamma*K(t) + ttheta*M(t)*B(t),
diff(M(t), t) = M(t) - ggamma*w(t)};

# Define the impulsive controls
impulse_controls := [1.0, 0.5, 0.8]; # Example impulsive control values

# Define the initial values and conditions
initial_values := [K(0) = 0, B(0) = 0, M(0) = 0];

# Define the final conditions
final_conditions := [K(T) = 0, M(T) = 0, B(T) > 0];

# Define the objective function to be maximized
objective := B(T) - int(c1*w(t)^2, t = 0 .. T) - add(c2*impulse_changes(ti), ti = 1.0 .. 3.0);

# Solve the system of differential equations numerically
sol := dsolve({diffeqs, initial_values, final_conditions}, numeric, output = listprocedure);

# Find the optimal control trajectory w(t) using optimization
w_optimal := optimize(objective, numeric, maximize);

# Evaluate the optimal control and state trajectories
optimal_controls := [seq(w_optimal(t), t = 0.0 .. T, 0.1)];
state_trajectories := [sol[2](t), sol[3](t), sol[4](t)];

optimal_controls, state_trajectories;

I am getting the error:

"Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations"

as soon I run after the sol:= function.

I would appreciate any help with fixing my code!

Thank you very much!

## How i can calculate invlaplace?...

How i can calculate invlaplace?

iman1.mw

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 (1)
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## how I can reach the result in dsolve...

Hi,

How can I find the same result reported in the figure for dsolve the differential equation?

nima.mw

## How can the surface common to two cylinders be dis...

1) the two cylinders are centered on the x and z axis respectively

2) any two intersecting cylinders

## How do I make these commands???...

Hi,

I want to define the functions 10 and 11 and then put them in the eq equation, then simplify them and get the unknown values after the solve command, but there are error.

And value the function psi ?

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 (3)

## how do I want to produce a graph that has differen...

the graf that I want to generate is like this one

## Windows 11 and Maple 2020...

Hei

Vet noen om Windows 11 støtter Maple 2020? Eller støttes det bare av Windows 10.

## seq slower than proc ...

I learned about Dodgson calculation of the determinant only recently (https://en.m.wikipedia.org/wiki/Dodgson_condensation).
I am only interested in symbolic expressions of the determinant.
Furthermore, I compared several methods. Not surprisingly, the build in method is the fastest. But why is the seq method slower than the proc method for the Dodgson method? Is there anything I could do to program it more efficiently?

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 > Dodgson := proc(M::Matrix)  MatrixDet(M); InnerMatrix(M) ^~ (-1) *~ MatrixDet(MatrixDet(M)); do if 1 < RowDimension(%) then InnerMatrix(%%) ^~ (-1) *~ MatrixDet(%); end if; until RowDimension(%) = 1; Trace(%): end proc:
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## Help using URL:-Get...

Just need some help using URL:-Get

Get("https://sdo.gsfc.nasa.gov/assets/img/browse/2023/05/05/20230505 _184918_512_0304.jpg")

I'm sure there are some tags to use but not sure how.  The site does show a script on best practices, but at the moment don't know how to apply them.  Can anyone offer some help?

## How can Maple display all of the possible triangul...

As an example, the second display in the web site below shows the 42 possible triangulations of a cyclic heptagon polygon.

https://en.wikipedia.org/wiki/Polygon_triangulation

## maple bug with sum...

Hello,

I am trying to evaluate this expression numerically or symbolically without success.

> sum(1/(4.0*n^2-4*n+4*100000000^2+1)/10^n,n=1..infinity);

Maple is having hard time to convert it to LerchPhi and even more to evalf(%);

PS : mathematica is doing it at any precision very fast or can translate this into hypegoemetric.

PS2 : I use maple 2020 on windows 10 64 bits.

## Can a colored shape in the xy plane be projected o...

Is there any simple way that the colored shape created in the xy plane by the uploaded code can be projected in the z direction onto the surface of the unit sphere centred at the origin?

Projection.mw