## expansion in a differential equations...

guys ,

in a differential equation i want to expand its variable, but i  have some problem with it for exponential term :jadid.mw

thanks guys

## Solving fractional differential equation...

How to find the determining equation for a system of fractional differential equation using Maple 15?

## Solving two or three simultaneous second order dif...

How do i proceed to solve two differential equations?

Two equations two unknowns is easy to solve in polynomial algebraic equations. Example: x+y=5; x-y=3; The solution is x=4; y=1 by adding the equations we arrive at.

The two equations are second order differential equations with two variables say temperature T (x,y) and velocity c(x,y). Assume any simple equation (one dimensional as well i.e. T(x) and c(x) which you can demonstrate with ease, I have not formulated the exact equations and boundary conditions yet for SI Engine simulation.

Ramakrishnan

## Calculate differential of commutators...

Hi,

I am trying to realize the following calculation in Maple.

$\left[\sum_{i=0}^n y_i(x) \partial_x^i , \sum_{j=0}^m z_j(x) \partial_x^j \right] \\ = \sum_{i=0}^n \sum_{j=0}^m \sum_{l=0}^i \binom il y_i(x) \left( \partial_x^{i-l} z_j(x)\right) \partial_x^{l+j} \\ - \sum_{j=0}^m \sum_{i=0}^n \sum_{l=0}^j \binom jl z_j(x) \left( \partial_x^{j-l} z_i(x)\right) \partial_x^{l+i} \ .$

Is there a way to make maple understand d/dx as a differential opperator and calculate with it? When i for example try to calculate diff(d/dx, x) it should give me d^2/dx^2 as a result. Unfortunately i don't know how to realize this.

Basic problem is i don't know how to realize operator expressions in maple like for example:

f(x) d/dx      ( f(x) is a smooth function of x here )

where when applied to a function h(x) it should result in f(x) d/dx h(x) .

Is that possible?

Thank you very much in advance.

## Vibrations with maple

Maple 18

In this section, we will consider several linear dynamical systems in which each mathematical model is a differential equation of second order with constant coefficients with initial conditions specifi ed in a time that we take as t = t0.

All in maple.

Vibraciones.mw

(in spanish)

Atte.

L.AraujoC.

## Introducing high order derivatives values in fixed...

Hi everybody!

It's nice to join in this forum.

I'm trying to get the analytic solution of the Bernouilli-Euler beam equation, with the next boundary conditions:

w(x,t) = displacements.

w(0,t) = 0   -> It's a cantilever beam. At the x=0 it's clamped.

diff(w(x,t),x) = 0.   -> the gyro in the clamp is zero.

E*I*diff(w(L,t),x,x) = 0  -> the moment at the end of the beam (x=L) is zero.

E*I*diff(w(L,t),x,x,x) = 0  -> the shear at the end of the beam (x=L) is zero too.

I'm not able to introduce the second and the third derivatives as boundary conditions in the pdsolve equation. I post the hole code:

restart;
ode := I*E*(diff(w(x, t), x, x, x, x))+m*(diff(w(x, t), t, t)) = 0;

s := pdsolve(ode, w(x, t));

ode1 := op([2, 1, 1], s);

ode2 := op([2, 1, 2], s);

f1 := op(4, rhs(ode1));

f2 := op(2, rhs(ode2));

sol1 := dsolve(ode1, f1);

sol2 := dsolve(ode2, f2);

sol := rhs(sol1)*rhs(sol2);

conds := [w(0, t) = 0, (D[1](w))(0, t) = 0, eval(I*E*(D[1, 1](w))(x, t), x = L) = 0, eval(I*E*(D[1, 1, 1](w))(x, t), x = L) = 0];

pde := [ode, conds];

pdsolve(pde, w(u, t));

And I get this error:

"Error, (in PDEtools/pdsolve) invalid input: pdsolve/sys expects its 1st argument, SYS, to be of type {list({<>, =, algebraic}), set({<>, =, algebraic})}, but received [I*E*(diff(diff(diff(diff(w(x, t), x), x), x), x))+m*(diff(diff(w(x, t), t), t)) = 0, [w(0, t) = 0, (D[1](w))(0, t) = 0, I*E*(D[1, 1](w))(L, t) = 0, I*E*(D[1, 1, 1](w))(L, t) = 0]]"

It's seems I'm introducing the Boundary conditions of the second and third derivatives in a wrong way, but I can't discover how to do it.

Thanks very much in advance to everybody!!

Ger89

P.D. - I have use this "tutorial" to write the code ( http://homepages.math.uic.edu/~jan/mcs494f02/Lec34/pde.html ). Thanks very much again.

## how to input 2nd order differential equation?...

Hello there, could someone tell me how to input this differential equation?

d2x/dt2 - (3)dx/dt + 2x = 2t -3

x=2, t=0, dx/dt=4

## Get values from numerical solving...

Hi there,

I'm trying to get the values from the output of a dsolve command.

I have a system of differential equations:

de1 := diff(V(t), t) = V(t)-(1/3)*V(t)^3-W(t)+Ie;
de2 := diff(W(t), t) = 0.8e-1*(V(t)+.7-.8*W(t))

For a range of the independent variable t and for some given values of the parameter Ie, I would like to get the value of the dependent variable V, as well as its minimum/maximum values for each Ie.

Can anybody suggest a solution please?

This is the worksheet: MaplePrimes_dsolve_min-max.mw

Thanks,

jon

## Solving a piecewise differential equation system...

Hi there,

I've got the following differential equation system:,

dU/dt = delta·dotD -lambda·U - kappa·U^2
dL/dt = (1-phi)·lambda·U + 1/4 ·kappa·U^2

being phi, delta, kappa, lambda, kappa some fixed parameters of the system, and where dotD (the derivative wrt time of a function D), which is defined a piecewise funtion:

dotD(t)=1/(3·T1)·DT for t in [0,T1]

dotD(t)=2/(3·(T2-T1-T))·DT for t in [T1+T,T2]

where T and DT are also known, and T1 approaches 0, and T2 approaches T1+T.

Setting the equation system in Maple and trying to solve it, gives a NULL result. However, trying to solve each piece separately seems to work fine.

Why is this?

Furthermore, taking limits for the [T1+T,T2] part (having solved each piece separately) yields an invalid limits point error. Ain't the possibility to take limits for both parameters at the same time?

Any ideas?

This is the Maple worksheet: MaplePrimes_LQ_model_solve.mw

Thank you.

jon

## January’s Featured Live Webinars

by:

Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on an upcoming webinars we think may be of interest to the MaplePrimes community.  For the complete list of upcoming webinars, visit our website.

Creating Questions in Maple T.A. – Part #2

This presentation is part of a series of webinars on creating questions in Maple T.A., Maplesoft’s testing and assessment system designed especially for courses involving mathematics. This webinar, which expands on the material offered in Part 1, focuses on using the Question Designer to create many standard types of questions. It will also introduce more advanced question types, such as sketch, free body diagrams, and mathematical formula.

The third and final webinar will wrap up the series with a demonstration of math apps and Maple-graded questions.

Clickable Calculus Series – Part #1: Differential Calculus

In this webinar, Dr. Lopez will apply the techniques of “Clickable Calculus” to standard calculations in Differential Calculus.

Clickable Calculus™, the idea of powerful mathematics delivered using very visual, interactive point-and-click methods, offers educators a new generation of teaching and learning techniques. Clickable Calculus introduces a better way of engaging students so that they fully understand the materials they are being taught. It responds to the most common complaint of faculty who integrate software into the classroom – time is spent teaching the tool, not the concepts.

## Analyzing a differential equation system with vary...

Hi there,
I have a set of differential equations whose solution, Jacobian matrix and its eigenvalues, direction field, phase portrait and nullclines, need to be computed.

Each of the equations has a varying parameter.

I know how to get the above for a single parameter value, but when I set a range of values for the parameters, Maple is not able to handle all cases as I would expect: solving the differential equation system:

eq1 := x*(1.6*(1-(1/100)*x)-phi*y)
eq2 := (x/(15+x)-0.3e-1*x-.4)*y+.6+theta
desys := [eq1, eq2];
vars := [x, y];

Error, (in unknown) invalid input: Utilities:-SetEquations expects its 2nd argument, equations, to be of type set({boolean, algebraic, relation}), but received {-600*y+(Array(1..2, {(1) = 8400, (2) = 15900})), Array(1..5, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 0})}

The equations are the following:
de1 := diff(x(t), t) = x(t)*(1.6*(1-(1/100)*x(t))-phi*y(t));
de2 := diff(y(t), t) = (x(t)/(15+x(t))-0.3e-1*x(t)-.4)*y(t)+.6+theta

the parameters being:
phi:=[0 0.5 1 1.5 2]
theta:=[5. 10.]

How can I handle the situation so that Maple computes each of the above for each combination of the parameters?

I would like to avoid using two for loops and having to store all results in increasingly bigger and complicated arrays.

The worksheet at issue is this: MaplePrimes_Tumour_model_phi_theta_variation.mw

Thanks,
jon

## abs function dsolve...

Hi all,

I am trying to solve the following differential equation numerically using dsolve,

y * abs (y''') = -1

y(0) =1, y'(0) = 0, y''(0)=0

it works fine when tthe absolute function is not there, i.e. yy''' = -1.

Do you have any suggestion?

## Compute and display isoclines (nullclines) for an ...

Hi there,

I would like to compute and display the nullclines of a set of ordinary differential equations.

AFAIK, I can compute the nullclines in Maple by defining the equations and solving the system

e.g.:

# Define the equations
eq1 := u(t)*(1-u(t)/kappa)-u(t)*v(t) = 0;
eq2 := g*(u(t)-1)*v(t) = 0;

# Solve the system (i.e. compute the nullclines)
sol := solve({eq1, eq2}, {u(t), v(t)});

However, I am not quite able to imagine how to display them over a dfieldplot or a phaseportrait.

Attached is an example with some differential equations, and their vector field and trajectories: MaplePrimes_Predator_prey_model_nullclines.mw.

It can be use to illustrate how to (compute and) display the nullclines.

Thank you,

jon

## Solving non-linear ODE...

Hi,

I'm trying to solve the following non-linear ODE numerically:

by ececuting

but maple gives me this error-message:

"Error, (in dsolve/numeric/make_proc) Could not convert to an explicit first order system due to 'RootOf'"

I couldnt find any useful information in the manual. What does this error mean? Is there something wrong with my maple code or is there just no solution for this particulare differential equation?

## Substituting an expression into an expression...

I'm trying to substitute one Differential equation into another differential equation.

eq1:=d*n(t)/dt = (rho(t)-beta)*n(t)/Lambda+lambda*C(t)+q

eq2:=diff(eq1, t)

resulting in -> eq2 := d*(diff(n(t), t))/dt = (diff(rho(t), t))*n(t)/Lambda+(rho(t)-beta)*(diff(n(t), t))/Lambda+lambda*(diff(C(t), t))

then I'm given that (diff(C(t), t)) is given by another equation:

eq3:=d*C(t)/dt = beta*n(t)/Lambda-lambda*C(t)

At this point I'm trying to substitute equation 3 into equation 2 for diff(C(t),t)

eq4 := subs(diff(C(t), t) = rhs(eq2), eq5)

however no matter what way's I try this I get an error:

Error, (in simpl/reloprod) invalid terms in product: (d*(diff(n(t), t))/dt = (diff(rho(t), t))*n(t)/Lambda+(rho(t)-beta)*(diff(n(t), t))/Lambda+lambda*(diff(C(t), t)))^-1

I then tried to map it but again i got an error specifically about the first parameter:

Error, invalid operator parameter name

eq5:=map((d/dt C(t))->beta/Lambda*n(t)-lambda*C(t),eq2)

I'm just wondering if what I am trying to do is even possible in Maple?

If anyone can help I would greatly appreciate it!

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