Apply combine(expand... to the right-hand side of the solution given by Maple's dsolve command. The result will be exactly the same as the "hand" solution.

My attention was drawn to this example because of past experience with Maple's use of variation-of-parameters to find a particular solution to a linear, constant-coefficient ODE. In the past, Maple did not check that the particular solution might contain constant multiples of members of the fundamental set. This caused a great deal of grief in my Advanced Engineering Mathematics with Maple ebook, and I was interested in discovering if that was the issue here. Apparently, it's not.

RJL Maplesoft

collect(expand(f2),diff,distributed)

The result will be similar to what Acer's code gives.

RJL Maplesoft

The task template at Tools/Tasks/Browse-Differential Equations/ODEs/Phase Portrait - Autonomous Systems will create a phase portrait interactively. Enter the ODEs, the critical points, ranges for the window and the parameter along the orbits. Then press the Enter Data button. On the resulting direction field, click on a point through which an orbit is desired. A full phase portrait can be built up without the need to tediously enter each initial point into the phaseportrait command.

RJL Maplesoft

interface(imaginaryunit = j)

This frees the default I=sqrt(-1) and sets j = sqrt(-1). I can now be assigned to, and used as an ordinary name.

RJL Maplesoft

Mathematically, the problem presented is this: Given the function psi(x,eta) where eta is itself a function of x and y, obtain the partial derivatives of psi with respect to x and y. The notation for this is psi(x,eta(x,y)). Hence, the derivatives are diff(psi,x)+diff(psi,eta)*diff(eta,x) and diff(psi,eta)*diff(eta,y).

Unfortunately, implementing those commands in Maple won't work once eta is defined because psi will inherit that definition for the "eta" used in defining psi. One way around that is to define psi with x and eta, and define Eta(x,y) instead of eta(x,y). Then, change diff(eta...) to diff(Eta...). One can always replace eta by Eta in the resulting answers to see the form of the solution obtained by Preben.

Preben's method takes less coding in this case, but could become more cumbersome in more complex cases where an explicit display of the functional dependencies might just be too messy.

RJL Maplesoft

Not all the Large Operators are connected to the Maple language. In particular, none of the integral symbols are anything more than notation.

RJL Maplesoft

The Student Calculus package has the MeanValueTheorem command, and a tutor that implements the command. The question asked had to do with applying constraints on a solution, and the answer given shows how to do this. But if the built-in command would have sufficed, it would have obviated the need to write a procedure.

Also, I wonder why the limit-definition of the derivative was used to obtain derivatives. There's a built-in diff command for that.

RJL Maplesoft

I added the option "output=integral" to see just how difficult this integral might be. The integrand is pretty complicated. I didn't have patience to wait for the calculation to produce the message about running out of memory. I doubt the integral can be evaluated in closed form. If the parameter r is given a numeric value, the integral can be evaluated numerically instantly. Not much help, though, if a closed-form is essential.

RJL Maplesoft

Commands in the Optimization package should support infolevel. See the help page for the Optimization package itself.

RJL Maplesoft

Look at the help page for try, a mechanism for detecting an error condition, with provision for what Maple code to execute upon finding the error.

RJL Maplesoft

Not completely sure what you want, but consider the following constructions.

q1:=polar(3,2.):
q2:=polar(5,1.5):
C1:=evalc(q1):
C2:=evalc(q2):
V1:=<Re(C1),Im(C1)>:
V2:=<Re(C2),Im(C2)>:
V2:=VectorCalculus:-RootedVector(root=V1,V2):
VectorCalculus:-PlotVector([V1,V2],color=[black,red],scaling=constrained,width=.5);

The "phasors" would be q1 and q2, complex numbers expressed in terms of magnitude and angle. C1 and C2 would be the rectangular forms for q1 and q2. V1 and V2 are then planar vectors representing these complex numbers, but V2 has its tail at the head of V1. (This construction is the simplest way I know how to put the tail of a vector at a specific location.) The PlotVector command is the nicest way I know to graph several vectors at once.

RJL Maplesoft

The curve you proposed is an elliptic helix if a does not equal b.

If you load the Student VectorCalculus package, and write the curve as

<X,Y,Z>, where X,Y,Z are the appropriate functions of t, and right-click on this, in the Context Menu that appears you will see Student Vector Calculus, and in that, Tutors. Pick the Space Curves tutor.

The tutor that launches will have your vector in it, and you can modify the domain for t. You get a graph of the curve, and you can also ask the tutor to show the various Frenet-Serret basis vectors, even animating them to move along the curve.

RJL Maplesoft

SurfaceOfRevolution(x^2, 1..2, 'axis'='horizontal', 'distancefromaxis' = 1, 'output'='plot');

This command lives in the Student Calculus1 package. I obtained this exact syntax by launching the SurfaceOfRevolution tutor, using it to craft the requisite surface, then copied (and pasted) the command from the bottom of the tutor.

RJL Maplesoft

Ages ago I asked one of our developers this question. Here is the reply I have recorded in my Little Red Book of Maple Magic.

Q:=indets(indets(ode,'specfunc'('anything','diff')),'And'('function','Not'('known')));

Q will be of the form {y(x)} and op(0,Q[1]) gives y, while op(1,Q[1]) gives x.

I suspect the single quotes are a habit of our developers who do their best to protect users from themselves.

RJL Maplesoft

If I understand your question correctly, you need to apply the differentiation operator d/du to F(x(u),y(u)). Otherwise, your ambiguous use of "total derivative" blows by me like a knuckle-ball. (A baseball allusion: a knuckle-ball is a spinless pitch for which air resistance causes erratic fluctuations that baffle most batters, and even the catcher.)

RJL Maplesoft