## How to achieve its numerical solution？...

I  encountered a non-integrable integral in the process of solving the following process, . How to achieve its numerical solution? Such as in a looping   code：

#######
pa[i] := pa[i-1]-(Int(subs(t = tau, Lpa[i-1]+Na1[i-1]-Na2[i-1]), tau = 0 .. t));

pw[i] := pw[i-1]-(Int(subs(t = tau, Lpw[i-1]+Nw1[i-1]-Nw2[i-1]), tau = 0 .. t)); u[i] := u[i-1]-(Int(subs(t = tau, Lu[i-1]+Nu1[i-1]+Nu2[i-1]), tau = 0 .. t))；

######
Detailed code see annexBC2.mw

## surd produces absurd result...

I am trying to solve an equation using surd and I get a strange result.

solve(surd(x^4,8)=-2)
4, -4, 4 I, -4 I

These solutions are clearly wrong.

The equation (x^4)^(1/8) = -2 has no solution.

This problem is equivalent to asking the computer to solve sqrt(x) = -2

which has no solution in R or C.

However if I type

solve((x^4)^(1/8) = -2) , then I get no answer, which is what I expected.

Why does surd behave in this unexpected way.

Also another thing I am wondering, why doesn't Maple simplify (x^4)^(1/8) to x^(1/2).

I tried the simplify command it didn't work.

## With(rootfinding) Isolate help!!...

I am having 26th degree polynomial univariate equation , I used Isolate to get the roots. but I am getting some extra roots which are not true they I even tried to substitute those roots in original equation then I got non zero answer instead of getting nearly zero answer.How is it possible??

equation looks like:

-12116320194738194778134937600000000*t^26+167589596741213731838990745600000000*t^24+1058345691529498270472972795904000000*t^22-4276605572538658673086219419648000000*t^20-23240154739806540070988490473472000000*t^18-5442849111209103187871341215744000000*t^16+49009931453396028716875310432256000000*t^14+74247033158233643322704589225984000000*t^12-2762178990802317464801412907008000000*t^10-25947900993773120244883450232832000000*t^8-7468990043547273070742668836864000000*t^6-567730116675454293925108383744000000*t^4+3703566799705707258760396800000000*t^2-4742330812072533924249600000000

Solutions i got:

[t = -4.162501845, t = -2.295186769, t = -1.300314688, t = -.8048430445, t = -0.6596008501e-1, t = -0.4212510777e-1, t = 0.4212510777e-1, t = 0.6596008501e-1, t = .8048430445, t = 1.300314688, t = 2.295186769, t = 4.162501845]

t=4.162501845 give me non zero answer when I substitute it in the equation given above:

## PDE solutions: when are they "general"?

by: Maple

Hi,
The latest update to the differential equations Maple libraries (this week, can be downloaded from the Maplesoft R&D webpage for Differential Equations and Mathematical functions) includes new functionality in pdsolve, regarding whether the solution for a PDE or PDE system is or not a general solution.

In brief, a general solution of a PDE in 1 unknown, that has differential order N, and where the unknown depends on M independent variables, involves N arbitrary functions of M-1 arguments. It is not entirely evident how to extend this definition in the case of a coupled, possibly nonlinear PDE system. However, using differential algebra techniques (automatically used by pdsolve when tackling a PDE system), that extension to define a general solution for a DE system is possible, and also when the system involves ODEs and PDEs, and/or algebraic (that is, non-differential) equations, and/or inequations of the form  involving the unknowns, and all of this in the presence of mathematical functions (based on the use of Maple's PDEtools:-dpolyform). This is a very nice case were many different advanced developments come together to naturally solve a problem that otherwise would be rather difficult.

The issues at the center of this Maple development/post are then:

a) How do you know whether a PDE or PDE system solution returned is a general solution?

b) How could you indicate to pdsolve that you are only interested in a general PDE or PDE system solution?

The answer to a) is now always present in the last line of the userinfo. So input infolevel[pdsolve] := 3 before calling pdsolve, and check what the last line of the userinfo displayed tells.

The answer to b) is a new option, generalsolution, implemented in pdsolve so that it either returns a general solution or otherwise it returns NULL. If you do not use this new option, then pdsolve works as always: first it tries to compute a general solution and if it fails in doing that it tries to compute a particular solution by separating the variables in different ways, or computing a traveling wave solution or etc. (a number of other well known methods).

The examples that follow are from the help page pdsolve,system, and show both the new userinfo telling whether the solution returned is a general one and the option generalsolution at work.The examples are all of differential equation systems but the same userinfos and generalsolution option work as well in the case of a single PDE.

Example 1.

Solve the determining PDE system for the infinitesimals of the symmetry generator of example 11 from Kamke's book . Tell whether the solution computed is or not a general solution.

 >
 (1.1)

The PDE system satisfied by the symmetries of Kamke's ODE example number 11 is

 >

This is a second order linear PDE system, with two unknowns  and four equations. Its general solution is given by the following, where we now can tell that the solution is a general one by reading the last line of the userinfo. Note that because the system is overdetermined, a general solution in this case does not involve any arbitrary function

 >
 -> Solving ordering for the dependent variables of the PDE system: [xi(x,y), eta(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) <- Returning a *general* solution
 (1.2)

Next we indicate to pdsolve that  and  are parameters of the problem, and that we want a solution for , making more difficult to identify by eye whether the solution returned is or not a general one. Again the last line of the userinfo tells that pdsolve's solution is indeed a general one

 >
 (1.3)
 >
 -> Solving ordering for the dependent variables of the PDE system: [r, n, xi(x,y), eta(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to r tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to n tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) tackling triangularized subsystem with respect to xi(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to eta(x,y) <- Returning a *general* solution
 (1.4)
 >
 (1.5)

Example 2.

Compute the solution of the following (linear) overdetermined system involving two PDEs, three unknown functions, one of which depends on 2 variables and the other two depend on only 1 variable.

 >

The solution for the unknowns G, H, is given by the following expression, were again determining whether this solution, that depends on 3 arbitrary functions, , is or not a general solution, is non-obvious.

 >
 -> Solving ordering for the dependent variables of the PDE system: [F(r,s), H(r), G(s)] -> Solving ordering for the independent variables (can be changed using the ivars option): [r, s] tackling triangularized subsystem with respect to F(r,s) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... differential factorization successful. First set of solution methods successful tackling triangularized subsystem with respect to H(r) tackling triangularized subsystem with respect to G(s) <- Returning a *general* solution
 (1.6)
 >
 (1.7)

Example 3.

Compute the solution of the following nonlinear system, consisting of Burger's equation and a possible potential.

 >

We see that in this case the solution returned is not a general solution but two particular ones; again the information is in the last line of the userinfo displayed

 >
 -> Solving ordering for the dependent variables of the PDE system: [v(x,t), u(x,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, t] tackling triangularized subsystem with respect to v(x,t) tackling triangularized subsystem with respect to u(x,t) First set of solution methods (general or quasi general solution) Second set of solution methods (complete solutions) Trying methods for second order PDEs Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Trying methods for second order linear PDEs Preparing a solution HINT ... Trying HINT = _F1(x)*_F2(t) Fourth set of solution methods Preparing a solution HINT ... Trying HINT = _F1(x)+_F2(t) Trying travelling wave solutions as power series in tanh ... * Using tau = tanh(t*C[2]+x*C[1]+C[0]) * Equivalent ODE system: {C[1]^2*(tau^2-1)^2*diff(diff(u(tau),tau),tau)+(2*C[1]^2*(tau^2-1)*tau+2*u(tau)*C[1]*(tau^2-1)+C[2]*(tau^2-1))*diff(u(tau),tau)} * Ordering for functions: [u(tau)] * Cases for the upper bounds: [[n[1] = 1]] * Power series solution [1]: {u(tau) = tau*A[1,1]+A[1,0]} * Solution [1] for {A[i, j], C[k]}: [[A[1,1] = 0], [A[1,0] = -1/2*C[2]/C[1], A[1,1] = -C[1]]] travelling wave solutions successful. tackling triangularized subsystem with respect to v(x,t) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... Trying methods for PDEs "missing the dependent variable" ... Second set of solution methods (complete solutions) Trying methods for second order PDEs Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Trying methods for second order linear PDEs Preparing a solution HINT ... Trying HINT = _F1(x)*_F2(t) Third set of solution methods successful tackling triangularized subsystem with respect to u(x,t) <- Returning a solution that *is not the most general one*
 (1.8)
 >
 (1.9)

This example is also good for illustrating the other related new feature: one can now request to pdsolve to only compute a general solution (it will return NULL if it cannot achieve that). Turn OFF userinfos and try with this example

 >

This returns NULL:

 >

Example 4.

Another where the solution returned is particular, this time for a linear system, conformed by 38 PDEs, also from differential equation symmetry analysis

 >

There are 38 coupled equations

 >
 (1.10)

When requesting a general solution pdsolve returns NULL:

 >

A solution that is not a general one, is however computed by default if calling pdsolve without the generalsolution option. In this case again the last line of the userinfo tells that the solution returned is not a general solution

 >
 (1.11)
 >
 -> Solving ordering for the dependent variables of the PDE system: [eta[1](x,y,z,t,u), xi[1](x,y,z,t,u), xi[2](x,y,z,t,u), xi[3](x,y,z,t,u), xi[4](x,y,z,t,u)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u] tackling triangularized subsystem with respect to eta[1](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(x,y,z,t), _F2(x,y,z,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, y, z, u] tackling triangularized subsystem with respect to _F1(x,y,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F3(x,y,z), _F4(x,y,z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y, z, t] tackling triangularized subsystem with respect to _F3(x,y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F4(x,y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F5(y,z), _F6(y,z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [y, z, x] tackling triangularized subsystem with respect to _F5(y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F6(y,z) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F7(z), _F8(z)] -> Solving ordering for the independent variables (can be changed using the ivars option): [z, y] tackling triangularized subsystem with respect to _F7(z) tackling triangularized subsystem with respect to _F8(z) tackling triangularized subsystem with respect to _F2(x,y,z,t) First set of solution methods (general or quasi general solution) Trying differential factorization for linear PDEs ... Trying methods for PDEs "missing the dependent variable" ... Second set of solution methods (complete solutions) Third set of solution methods (simple HINTs for separating variables) PDE linear in highest derivatives - trying a separation of variables by * HINT = * Fourth set of solution methods Preparing a solution HINT ... Trying HINT = _F3(x)*_F4(y)*_F5(z)*_F6(t) Third set of solution methods successful tackling triangularized subsystem with respect to xi[1](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(x,z,t), _F2(x,z,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z, y] tackling triangularized subsystem with respect to _F1(x,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F2(x,z,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F3(x,t), _F4(x,t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, x, z] tackling triangularized subsystem with respect to _F3(x,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to _F4(x,t) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F5(x), _F6(x)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, t] tackling triangularized subsystem with respect to _F5(x) tackling triangularized subsystem with respect to _F6(x) tackling triangularized subsystem with respect to xi[2](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful -> Solving ordering for the dependent variables of the PDE system: [_F1(t), _F2(t)] -> Solving ordering for the independent variables (can be changed using the ivars option): [t, z] tackling triangularized subsystem with respect to _F1(t) tackling triangularized subsystem with respect to _F2(t) tackling triangularized subsystem with respect to xi[3](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful tackling triangularized subsystem with respect to xi[4](x,y,z,t,u) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. First set of solution methods successful <- Returning a solution that *is not the most general one*
 (1.12)
 >
 (1.13)

Example 5.

Finally, the new userinfos also tell whether a solution is or not a general solution when working with PDEs that involve anticommutative variables  set using the Physics  package

 >
 (1.14)

Set first  and  as suffixes for variables of type/anticommutative  (see Setup )

 >
 (1.15)

A PDE system example with two unknown anticommutative functions of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen let's use PDEtools:-diff_table   PDEtools:-declare

 >
 (1.16)
 >
 (1.17)

Consider the system formed by these two PDEs (because of the q diff_table just defined, we can enter derivatives directly using the function's name indexed by the differentiation variables)

 >
 (1.18)
 >
 (1.19)

The solution returned for this system is indeed a general solution

 >
 -> Solving ordering for the dependent variables of the PDE system: [_F4(x,y), _F2(x,y), _F3(x,y)] -> Solving ordering for the independent variables (can be changed using the ivars option): [x, y] tackling triangularized subsystem with respect to _F4(x,y) tackling triangularized subsystem with respect to _F2(x,y) tackling triangularized subsystem with respect to _F3(x,y) First set of solution methods (general or quasi general solution) Trying simple case of a single derivative. HINT = _F6(x)+_F5(y) Trying HINT = _F6(x)+_F5(y) HINT is successful First set of solution methods successful <- Returning a *general* solution
 (1.20)
 >

This solution involves an anticommutative constant , analogous to the commutative constants  where n is an integer.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## Difficulty in Solving a Problem...

I have been working on a general solution to motion analysis and seem to be going backwards.  I have an numerical solution in Octave I use for comparison.  I have reduced the problem to a small example that exhibits the problem.

I posted a question similar to this, but, without a set of known values.

I am doing something wrong, but, what?

Tom Dean

## bearing.mpl, solve the target motion problem with bearings only.
##
## Consider a sensor platform moving through points (x,y) at times
## t[1..4] with the target bearings, Brg[1..4] taken at times t[1..4]
## with the target proceeding along a constant course and speed.
##
## time t, bearing line slope m, sensor position (x,y) are known
## values.
##
## Since this is a generated problem the target position at time t is
## provided to compare with the results.
##
#########################################################################
##
restart;
##
genKnownValues := proc()
description "set the known values",
"t - relative time",
"x - sensor x location at time t[i]",
"y - sensor y location at time t[i]",
"m - slope of the bearing lines at time t[i]",
"tgtPosit - target position at time t[i]";
global t, m, x, y, tgtPosit;
local dt, Cse, Spd, Brg, A, B, C, R, X;
local tgtX, tgtY, tgtRange, tgtCse, tgtSpd;
## relative and delta time
t := [0, 1+1/2, 3, 3+1/2];
dt := [0, seq(t[idx]-t[idx-1],idx=2..4)];
## sensor motion
Cse := [90, 90, 90, 50] *~ Pi/180; ## true heading
Spd := [15, 15, 15, 22];  ## knots
## bearings to the target at time t
Brg := [10, 358, 340, 330] *~ (Pi/180);
## slope of the bearing lines
m:=map(tan,Brg);
## calculate the sensor position vs time
x := ListTools[PartialSums](dt *~ Spd *~ map(cos, Cse));
y := ListTools[PartialSums](dt *~ Spd *~ map(sin, Cse));
## target values  start the target at a known (x,y) position at a
## constant course and speed
tgtRange := 95+25/32; ## miles at t1, match octave value...
tgtCse := 170 * Pi/180; ## course
tgtSpd := 10; ## knots
tgtX := tgtRange*cos(Brg[1]);
tgtX := tgtX +~ ListTools[PartialSums](dt *~ tgtSpd *~ cos(tgtCse));
tgtY := tgtRange*sin(Brg[1]);
tgtY := tgtY +~ ListTools[PartialSums](dt *~ tgtSpd *~ sin(tgtCse));
## return target position vs time as a matrix
tgtPosit:=Matrix(4,2,[seq([tgtX[idx],tgtY[idx]],idx=1..4)]);
end proc:
##
#########################################################################
## t[], m[], x[], and y[] are known values
##
## equation of the bearing lines
eq1 := tgtY[1] - y[1]    = m[1]*(tgtX[1]-x[1]):
eq2 := tgtY[2] - y[2]    = m[2]*(tgtX[2]-x[2]):
eq3 := tgtY[3] - y[3]    = m[3]*(tgtX[3]-x[3]):
eq4 := tgtY[4] - y[4]    = m[4]*(tgtX[4]-x[4]):
## target X motion along the target line
eq5 := tgtX[2] - tgtX[1] = tgtVx*(t[2]-t[1]):
eq6 := tgtX[3] - tgtX[2] = tgtVx*(t[3]-t[2]):
eq7 := tgtX[4] - tgtX[3] = tgtVx*(t[4]-t[3]):
## target Y motion along the target line
eq8 := tgtY[2] - tgtY[1] = tgtVy*(t[2]-t[1]):
eq9 := tgtY[3] - tgtY[2] = tgtVy*(t[3]-t[2]):
eq10:= tgtY[4] - tgtY[3] = tgtVy*(t[4]-t[3]):
##
#########################################################################
##
## solve the equations
eqs  := {eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10}:

Sol:= solve(eqs, {tgtVx, tgtVy, seq([tgtX[k], tgtY[k]][], k= 1..4)}):
##

genKnownValues():
## these values are very close to Octave
evalf(t);evalf(m);evalf(x);evalf(y);evalf(tgtPosit);
## The value of tgtX[] and tgtY[] should equal the respective tgtPosit values
seq(evalf(eval([tgtX[idx],tgtY[idx]], Sol)),idx=1..4);

## Subscript error...

Hello

I have a subscripts error, or it seems to be an error.

As you can see on the picture, then I have defined the varible I__K, but when I need it again I get another result or It seems to be another result that looks like this I[K]. I use the esc buttom to recall the varible.

Are there anybody that has a solution to this? I have been looking at other treads, but there seems not to be a solution that works or maybe I'm looking the wrong places.

Regards

Heide

## Solutions lost by isolve command ...

I have the system:

Update:

{-1/2 < 2*f*(1/53)+7*g*(1/53), 3/106 < 7*f*(1/53)-2*g*(1/53), 2*f*(1/53)+7*g*(1/53) < -37/106, 7*f*(1/53)-2*g*(1/53) < 1/2}

which I wish to solve over integers but isolve() gives me "Warning, solutions may have been lost and no solutions". The solutions exist and are {[f =0, g = -3] || [f = 1, g = -4], [f = 1, g = -3] || [f = 2, g = -4]}, but I cannot obtain them with Maple. Could you tell me what is wrong and how I should treat this kind of problems in the future, please.

Mathematica 10.0

Reduce[{-1/2 < 2*f*(1/53) + 7*g*(1/53), 3/106 < 7*f*(1/53) - 2*g*(1/53), 2*f*(1/53) + 7*g*(1/53) < -37/106, 7*f*(1/53) - 2*g*(1/53) < 1/2}, {f, g}, Integers]

(f == 0 && g == -3) || (f == 1 && g == -4) || (f == 1 &&
g == -3) || (f == 2 && g == -4)

Maple

isolve({-1/2 < 2*f*(1/53)+7*g*(1/53), 3/106 < 7*f*(1/53)-2*g*(1/53), 2*f*(1/53)+7*g*(1/53) < -37/106, 7*f*(1/53)-2*g*(1/53) < 1/2});
Warning, solutions may have been lost

## Help...for component embedeed...

The solution should show the procedure.
But without tutor.

Problem.mw

Atte.

Lenin Araujo Castillo

## How to handle solution of ODE?...

in this code count.mw  i can get the value of solution only by typing sol(t), how to get the value of q(t) directly?

 >
 (1)
 >
 (2)
 >
 (3)
 >
 (4)
 >
 (5)
 >
 (6)
 >
 (7)
 >

## Lost solutions ...

Hi!

For some reason my three equation with three unknowns won't be solved. Anyone who can help? I tried the following:

Really appreciate anyone who can help me out!

## Optimization which uses solutions of system...

how to do optimization for two equations in terms of two variables

LPSolve({eq1}, {eq2}, assume = {nonnegative});

eq1 is a rational function and eq2 is a very large rational function

after run , it return error, objective function must be specified as a linear polynomial or vector

da := [LengthSplit(Flatten([[1,m],[2,m2],[seq([i+1,close3[i][1]], i=2..4)]]),2)];
f := PolynomialInterpolation(da, z):
solution := solve(f=-z, z, explicit);
zz := [x1,x2,x3,x4];
sigma := symMonomial(zz); #sigma := [x1+x2+x3+x4, x1*x2+x1*x3+x1*x4+x2*x3+x2*x4+x3*x4, x1*x2*x3+x1*x2*x4+x1*x3*x4+x2*x3*x4, x1*x2*x3*x4]
sys1 := subs([x1=solution[1],x2=solution[2],x3=solution[3],x4=solution[4]], sigma[1]):
sys2 := subs([x1=solution[1],x2=solution[2],x3=solution[3],x4=solution[4]], sigma[2]):
sys3 := subs([x1=solution[1],x2=solution[2],x3=solution[3],x4=solution[4]], sigma[3]):
sys4 := subs([x1=solution[1],x2=solution[2],x3=solution[3],x4=solution[4]], sigma[4]):
da := [seq([i,close3[i][1]], i=1..5)];

with(Optimization):
LPSolve(sys1, {sys2}, assume = {nonnegative});

## Equation concerning solutions of ODE system...

Hello all,

I have an ODE system (please see bellow) where my unknowns are S(t) and K(t), all the the other symbols are known parameters. This system, by given the initial values for S and K, that is, S(0)=100 and K(0)=20, I can solve numerically.

sys:= diff(S(t), t) = -eta*K(t)*S(t)/(w*N*(S(t)+K(t))), diff(K(t), t) = eta*K(t)*S(t)/(w*N*(S(t)+K(t)))+S(t)*(-z*eta*alpha*K(t)^2+(-z*eta*alpha*S(t)-(eta*alpha^2*S(t)^2-2*N*C[max]*w*eta*alpha*K(t)+((-N*w+z)*alpha+N*C[max]^2*w*eta)*w*N)*upsilon)*K(t)+N*S(t)*w*alpha*upsilon*(N*w-z))/((K(t)^2*alpha*z+3*K(t)*S(t)*alpha*z+(2*S(t)*z*alpha+upsilon)*S(t))*w*N)

In addition, I have an algebraic equation:

eq1:= -c2 + (K(t2)*S(t2)+w*N*S(t2))*z=0,

where S(t2) and K(t2) are the solutions of my ODE sys in S and K at t=t2. The t2 is unknown time variable.

My question is: how can I find t2 such that my algebraic equation (eq1) is satisfied.

Dmitry

## How to solve couple linear ODEs through power seri...

hi.how i can dsolve couple linear equations with power series solutions or taylor series expantion?

file attached below.

thanks

TAYLOR.mw

## How to find specific solutions?...

A lot of my life is at the moment spent using solve to solve systems of equations, and then trying to weed through the solutions maple gives to find the ones I am interested in. Specifically i'd like to have a program that can weed through the solutions and eliminate those that include equalities of the  form p[i]=-p[j] or p[i]=0  where i and j are integers (or equalities of that form with the letter q replacing p). Specifically i don't want to exclude equalities of the form p[i]=-p[j]*something+something else-another thing.... as they can be useful (or equalities of that form with the letter q replacing p).

Here is a (simple) example of the kind of equations I am likely to be solving and their output from solve:
A := solve([p[1]*p[2]*p[3] = q[1]*q[2]*q[3], p[1]+p[3] = q[1]+q[3], p[2]^2+p[3]^2 = q[2]^2+q[3]^2])

I have some code which gets rid of solutions where one variable is set to 0

with(ArrayTools);
GetRidOfDumbSolutions := proc (sols)
local Nsols, Npars, GoodSol, GoodSols, GoodSolsCounter, i, j;
Nsols := numelems(sols); Npars := numelems(sols[1]);
GoodSols := []; GoodSolsCounter := 0;
for i to Nsols do
GoodSol := 1;
for j to Npars do
if IsZero(rhs(sols[i, j]))
then GoodSol := 0
end if
end do;
if GoodSol = 1 then
GoodSols := Concatenate(1, GoodSols, sols[i])
end if
end do;
GoodSols
end proc

but i can't see how (in maple) to detect an expression of the form p[i]=-p[j] especiall if that is being written in 2-d math. (i don't quite understand the different maths environments or how to convert from one to another or to string)

## Compute values using function ...

Dear all;

Hello everybody, I need your help to dispaly some values obtained using my function f. When I run the code there is no results obtained. Many thanks.

restart:

# The vectors e(i) satify the folowing conditions
e(0)*e(1)=e(n-1) assuming  1<n;
e(0)*e(0)=e(2):
e(1)*e(1)=e(n-1) assuming  1<n: :
e(2)*e(1)=e(n) assuming  1<n:
#
for i from 1  to n-1 do
e(i)*e(0)=e(i+1);
end do:

# We define the function f
f:=e(0)->e(0)+(n-3)*e(1);
f:=e(1)->(n-2)*e(1);
for i from 2  to 3 do
f:=e(i)->(n+i-3)*e(i)+(i-1)*(n-3)*e(n-3+i);
end do:

for i from 4 to n do
f:=e(i)->(n+i-3)*e(i)
end do:

# We define the two vectors
x:=sum(alpha(k)*e(k),k=0..n);
y:=sum(beta(k)*e(k),k=0..n);

#Question : I would like to compute the following  but there is no display of the solution.
(x*y);
f(x*y);
f(x);
f(y);
x*f(y);
f(x)*y;
f(x*y)- f(x)*y-x*f(y);

 1 2 3 4 5 6 7 Last Page 1 of 11
﻿