There is something odd about Lesson 27 (Review of Matrices) as it appears in the course framework. In the section "Matrix Functions", the code snippet

A={{x^2, x},{3 + x}, Exp[x]}};
A // MatrixForm

is followed by (apparently output) matrix

{{{r^2, s^2}, {r, s}}, {{r + 3, s + 3}, {Exp[r], Exp[s]}}}  (* Completely different matrix *)

When I open the downloaded lesson in Mathematica, I see the input as shown, but the output is the matrix corresponding to the input with the variable x, not the more complicated matrix with r and s as I see in the course framework..

In Problem Session 7, the discussion of Problem 2 says "Because of the lack of a damping term, the solution takes an unending oscillatory path:"

In fact, the equation for this problem (y''[t] + 2*y'[t] + y[t] == (a few step functions) ) DOES have a damping term and the plotted solution tapers off quickly.

In Examples 3, 4, and 5 of Lesson 26 (Convolution Theorem), the analysis stops at the same point, showing that the solution has the form

Integrate[ someFunction[ t-r ]*phi[r], {r,0,t}]

where phi[r] represents the UnitStep function. Since the Wolfram Language apparently does not understand the meaning of phi, it does not resolve the integral . HOWEVER, it would help the student if you would point out in all these cases that the UnitStep function (phi) simply equals 1 in the region of integration {r, 0, t}, and therefore the integral reduces to Integrate[ someFunction [t-r], {r,0,t} ]. I'm sure it would also be useful to remind the student that L[UnitStep] is 1/s .

In Lesson 24, Exercise 4 also contains a sign error. The forcing function described at the top is

UnitStep[t-2*Pi] * Sin[t] - UnitStep[t-4*Pi] * Sin[t]

With its negative sign, the second term cancels out the first term after t==4Pi. The forcing function is then just a single cycle, and the solution settles down to a constant amplitude. However, the forcing function in the solution is UnitStep[t-2Pi] * Sin[t] + UnitStep[t-4*Pi] * Sin[t]

With its plus sign, the second term reinforces the sinusoidal oscillation driving the solution, which then diverges.

In the Exercises for Lesson 24 (Discontinuous Forcing Functions), Exercise 1 solves a different equation than posed at the top of the problem. At the top, the equation is

2*y''[t] + y'[t] +2*y[t] == UnitStep[t-2]   (* Note the sign of the y'[t] term. *)

The equation you actually solve for is

''[t] -y'[t] + 2*y[t] == UnitStep[t-2]      (* Note the sign of the y'[t] term. *)

Hello Richard,

I'll open a ticket including your report and notebook for Luke to review. I'll follow up with you via email.

Thank you,

Christine Owens
Wolfram U Project Manager
Wolfram U

Hello Jason,

I'll inquire about this with Luke.

Thank you,

Christine Owens
Wolfram U Project Manager
Wolfram U

Hello,

Thank you for bringing this to our attention. Can you please confirm your name? I'll follow up with you via email once this has been reviewed and updated.

Best,
Christine Owens
Wolfram U Project Manager
Wolfram U

Cool! Thank you. SUGGESTION: Add this to the discussion of Example 4 in Lesson 19. It seems to be a nice example of an apparent singularity that isn't "essential"; that is, it goes away when you look at it right.

Sorry to submit so many questions -- but not really. They come up because the course is so rewarding and has so much potential. Thanks for the challenge!

In Lesson 19 (Series Solutions near a Singular Point), Example 2, I'm able to duplicate everything up until the solution to the recursion relations (notebook attached). When I try to use RSolve[ ], I get expressions involving Pochhammer[...], which I've never heard of before. Wolfram documentation for Pochhammer indicates it should reduce to ratios of factorials if it is given integer arguments, but the solutions I come up with all involve a fraction.

QUESTION: When RSolve[ ] gives us Pochhammer, is there anything we can do to simplify it? Or is there something in particular we should look for to fix it?

Attachments:

Lesson_19_Exampl...nb

Thank you very much for all of your comments, Richard. I really do appreciate it. We've put a lot of work into this course and we want to make it the best course we can. Your comments have helped us a lot with identifying some of the errors or improvements that we've overlooked.

Hello Richard,

Thank you for this feedback. I'll create a ticket per your report for Luke to review.

Christine Owens Wolfram U Project Manager Wolfram U

The voiceover for Lesson 18 (Euler Equations) includes an important sentence that is NOT in the written notes: "The equation can be simplified by assuming the solution is some power of x." [At time stamp 2:09 or so.] The written notes then say "Euler observed that by utilizing the power rule (x^r}' == r*x^(r-1), the differential equation could be transformed into a simpler form ..."

SUGGESTION: Include the voiceover assumption that y[x]==x^r in the written notes. It's implicit in the Wolfram Language transformation ( /.y->Function[x,x^r] ) but has no justification if one is just reading the notes.

In Lesson 17 about series solutions near an ordinary point, the first graph shows a sequence of plots converging to a limit. The last graph shows a similar collection of plots, and points out their convergence in the interval (-1, 1). I was confused at first because I expected a note about divergence.

SUGGESTION: For the last graph, also point out that as the order increases the plots move AWAY from the full solution (dotted line) rather than toward it as they did in the first graph. This illustrates divergence outside the region of convergence.

Problems 11 through 14 of Problem Set 4 entail physical equations with actual physical units. The use of physical units such as {kilograms, meters, seconds} was mentioned in the class, so I tried using pounds, inches, and seconds, (converting pounds to mass by dividing by gravitational acceleration). Mathematica allowed me to enter pretty-looking equations and initial conditions, but then did not seem to know what to do with them when I tried DSolveValue. Presumably I'm doing something wrong, but the sample problems don't help here.

SUGGESTION: Demonstrate use of physical units in one or two problem set solutions.

Oops! Thanks for sharing this error in the survey question. We will fix it.

Hello Richard,

I've created a ticket per your inquiry for review.

Christine Owens
Wolfram U Project Manager
Wolfram U

I believe I figured out my original problem: The solution to Problem 5 refers to the Method of Undetermined Coefficients even when it is applying Variation of Parameters. Please check: Above the equation

u1=-Integrate[y2[x]Sin[x]/Wronskian[{y1[x],y2[x]},x],x]+C[1] 

do you mean to say "Method of Variation of Parameters" instead of "Method of Undetermined Coefficients"?

To form an augmented matrix from matrix A and vector b, it's important to realize that the augmented matrix uses b as a COLUMN vector. If b is a single-level list, that amounts to a row vector (in spite of what some Mathematica documentation says), so we must transpose it. Transpose[ ] works with matrices (multi-level lists) but not with row vectors (single-level lists). Assume A is {{1,2,3},{4,5,6}} and b is {x,y}. To transpose b we must first enclose it in curly braces to form a two-level list. We then direct the Join[ ...] function to act at level 2 of the lists, instead of level 1.

Augmented matrix:: Join[A,Transpose[ { b } ], 2]

Output: {{1, 2, 3, x}, {4, 5, 6, y}}

In both Lesson 24 (Discontinuous Forcing Functions) and Lesson 25 (Impulse Functions), plots are shown of the complementary solution (forcing function is zero) and the particular solution (forcing function/impulse function applied). It would be interesting to see a plot of the DIFFERENCE between results with/without forcing function.

In Lesson 24, there's not much to see. However, the difference plot of (sln2 - sln3) in Lesson 25 makes it easier to see the effects of the applied impulse functions, particular the first negative hit at t==2Pi. It's harder to see the effect of the positive hit at t==3Pi because it simply adds to the already-positive motion of the curve. SUGGESTION: Create a Manipulate box to allow the student to tinker with the timing and magnitude of the second hit. That should provide a nice feeling for coherent and destructive effects.

Thank you very much for your comments, Richard. Sorry that I couldn't give a better reply during the session today, but posting these comments here helps us keep track of everything that needs to be improved in the course.

Yes, you are absolutely right. It should be Integrate[ ( 0 + s^2) ^2, {s, 0, t}]. Thank you for pointing this out. We'll get it fixed.

That can't be controlled from within DSolveValue, but you can use the TrigToExp function to do that conversion. For example:

In[1]:= TrigToExp[DSolveValue[y''[x] + y[x] == x, y[x], x]]
Out[1]= x + 1/2 E^(-I x) C[1] + 1/2 E^(I x) C[1] + 1/2 I E^(-I x) C[2] - 1/2 I E^(I x) C[2]

Thank you very much. It does look like the solution for those two exercises have been switched. We'll have to fix that. Thank you for pointing that out. We really appreciate it.

Picard's Theorem works best when the initial condition when the dependent variable equals zero when the independent variable equals zero. Exercise 3 shows how to transform a differential equation where y[0] is not equal to zero into an equivalent problem where the initial condition at zero equals zero.

In Section 12 of the course, the symbol Lambda was used for the characteristic equation. Can you explain why that particular symbol was used there? It seems that choice was no accident, but I see no clues to why that was chosen. TY.

REQUEST: Can you please post a URL for the daily study group sessions? Everything I try just takes me to the registration page. It has dial-in information that doesn't work (no Dial-In ID Number), and I can't find a link to the session that has surely started by now.

Thanks, but that link goes to the course framework with the recording of the course. I'm looking for a link to the daily lectures as they are given. Sometimes the link in the daily reminder e-mail works, but at least once or twice it did not seem to work for me.

No, I'm NOT looking for a link to the recording. Cassidy has been sending those out each day. What I'm looking for is a link to the presentation ABOUT TO BE PRESENTED -- in other words, the live class session. For example, I just received the reminder for today's course session. It has a link [Join Daily Study Group] and immediately below that another link headed by "Join using your custom link:"

Both links take me to to a page describing the webinar, with a graphic that sometimes contains an arrowhead that lets me into the course and sometimes does not. Yesterday it did not. Today it does not. I have dialed into the session (Dial-In Number 176487# and Passcode 8751#) and am now listening to some very nice hold music ...

Richard, it sounds like you have an issue logging into the BigMarker servers. The link Cassidy sends out every day contains your BigMarker credentials: they're embedded in the long list of characters in that URL. Cassidy should be able to reset and re-send to you. An email to wolfram-u@wolfram.com should get the ball rolling.

As a temporary workaround, I bet you could "register" with another e-mail address and use that to access the session today. Wolfram U would prefer to fix your permanent account (since that's how they track users for certifications they earn), but that can get you running today.

Good luck. This course is hard enough for me with my BigMarker login working smoothly. :)

I'm trying to generate nice legends like the ones in the Autonomous Equations section of Lesson 6:

C[1] = -2  C[1] = -1  C[1] = 0  C[1] = 1  C[1] = 2 

I tried Table["C[1] = "j, {j, -2, 2}] but this generated

-2 C[1] =
-C[1] = 
0
C[1] =
2 C[1] =

In other words, Mathematica did math instead of lexical substitution!
QUESTION: Is there an analog of the C language sprintf function by which we can generate formatted expressions?

Something seems odd in the "Integrating Factor Method" section of Lesson 5 (Linear 1st-Order Eqns). The input

D[Exp[f[x]],x] 

generates Exp[f[x]] f'[x] on my system, as expected. However, in the lesson notebook it generates output with an exponent that seems to be a detailed list of possibilities. Pasting it into this box gives an unholy mess, but the first set of possibilities seems to be

1 if 0 <= x < 1 and
0 otherwise

and the second set of possibilities looks like

0 if x is not equal to zero or one;
Indeterminate otherwise

My guess is that the function f is carrying a definition over from a previous cell. SUGGESTION: clear the contents of f before calling D[Exp[f[x]],x]

In Lesson 5 (Linear 1st Order Eqns), the example for constant coefficients shows a manual reworking of the solution to bring the arbitrary constant C out of the exponential function, then uses that manually reworked expression as a replacement for y when checking the solution:

y->Function[x,(b+\[ConstantC] Exp[-a x])/a]

When I tried simply replacing y with the generated solution

[In]    soln2 = Solve[soln, y[x], Reals]

    (y'[x] + a*y[x] == b) /. y -> Function[x, Evaluate[soln2]] // Simplify

[Out]    {{y[x] -> (b + E^(-a x + C[1]))/a}}

the result would not simplify to True. It doesn't perform the indicated replacements:

[In]    (y'[x] + a*y[x] == b) /. y -> Function[x, Evaluate[soln2]] // Simplify

[Out]    
{{a (y[x] -> (b + E^(-a x + C[1]))/a) + (Derivative[1][y][x] -> -E^(-a x + C[1]))}} == b

QUESTION #1: Why won't it simplify?

QUESTION #2: Is there a way to have Mathematica do the work, rather than having to rework the solution manually?

When first studying differential equations, I seem to recall we were drawing magnitudes with the vectors in the direction field. Or maybe my memory is not accurage. I mostly recall they were a pain to draw manually -- much work. :( Is there any advantage to drawing magnitudes with the vector fields, or has everyone agreed that having unit magnitude vectors is the way to go?

Does VectorPlot[] allow one to draw vector fields with something other than a unit magnitude? Have you ever encountered an application where that was useful?

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Can r be i?

You can get an imaginary r(t) function, for example, by using an initial condition on r[0] that is complex. However, we are mostly focusing on the real-valued functions in this course.