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Wolfram U

[WSG77] Daily Study Group: Differential Equations (begins April 13)

Luke Titus, Wolfram Research
Posted 21 days ago

Please join us in a study group devoted to differential equations that begins Monday, April 13. This study group will meet daily, Monday to Friday, over the next two weeks. We will share the excellent lesson videos from the Wolfram U course "Introduction to Differential Equations." The study group sessions will include time for exercises, discussion and Q&A. This study group will help you achieve the course completion certificate for the "Introduction to Differential Equations" course after you complete the course quizzes.

REGISTER HERE

enter image description here
POSTED BY: Luke Titus
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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.
POSTED BY: Richard Daehler-Wilking

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.
POSTED BY: Richard Daehler-Wilking

Thanks, Richard. That's a good suggestion. We can include units in a few of those problems to show how to include units in a calculation.

POSTED BY: Luke Titus
Updating Name
Updating Name
Posted 2 days ago
POSTED BY: Updating Name

Thank you very much. It does look like it should be Exp[x] rather than Exp[2]. We'll get that fixed.

POSTED BY: Luke Titus
Rich Albert
Rich Albert
Posted 2 days ago

Hi Luke,

I decided to go ahead and work the quizzes for the certification, and on the second quiz regarding whether all autonomous eqtns are exact (the last question), my answer was marked as incorrect, whereas I am confident that it was correct. Is there some way that it can be reviewed by a person instead of AI so that proper credit can be given?

Thanks!
POSTED BY: Rich Albert

Hi Rich. Please send your solution to wolfram-u@wolfram.com They will forward your solution along to me and I will get back to you over email.

POSTED BY: Luke Titus

I have a similar problem with the free-response question on Quiz 4. I'll do as you suggest.
POSTED BY: Richard Daehler-Wilking

COMMENT on END-OF-COURSE SURVEY: You ask us to indicate all the Daily Study Groups we'd be interested in, but the radio buttons only allow a single choice. SUGGESTION: Change the reply function to checkboxes.
POSTED BY: Richard Daehler-Wilking

Oops! Thanks for sharing this error in the survey question. We will fix it.
POSTED BY: Jamie Peterson
POSTED BY: Richard Daehler-Wilking

Hello Richard,

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

Christine Owens
Wolfram U Project Manager
Wolfram U
POSTED BY: Christine Owens

Thanks for pointing that out, Richard. We'll get that fixed.

POSTED BY: Luke Titus
Updating Name
Updating Name
Posted 2 days ago
POSTED BY: Updating Name

Hello Richard,

I've added your follow-up to our original ticket per your report above for "problem 5, problem session 4".

Thank you,

Christine
POSTED BY: Christine Owens
POSTED BY: Richard Daehler-Wilking

Thank you, Richard. It looks like that is doing the right thing.

POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking
POSTED BY: Christine Owens

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.

POSTED BY: Luke Titus

I think there's a significant typo in the solution to Problem 11 of Problem Set 2. The problem is to plot the first four iterations of Picard's method for this initial value problem:

y'[t] == (t^2 + y[t]^2 )^2      and      y[0] == 0

The published solution starts out with 

y1 = Integrate[0 + s^2, {s, 0, t}]

This omits the outer exponent of the original equation. Shouldn't it be

y1 = Integrate[ ( 0 + s^2) ^2, {s, 0, t}]
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking

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]
POSTED BY: Luke Titus

I have notice between the Exercise 11-1 and Exrcise 12-1. They are changed.

enter image description hereenter image description here
POSTED BY: Jaime Jiménez

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.

POSTED BY: Luke Titus

I must have missed something. Lesson 8 is on Picard's Theorem. The first two exercises for Lesson 8 are indeed connected to Picard's Theorem, but exercises 8.3 through 8.5 seem unrelated to it. For example, Exercise 8.3 is:

Transform the following initial value problem to the equivalent problem with the initial value at the origin: 2x - y[x] + y'[x](2*y[x] - x) == 0 and y[1] == 5

The solution doesn't make sense to me, and Mathematica says the solution's transformed equation does NOT correspond to the original. The original solution is entirely real; the transformed solution is complex. Perhaps I made a typo, but I triple-checked it and also used copy-and-paste.
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus

In Lesson 7 / Exercises, the solution to Exercise 1 seems to be a non sequitur. The equation is 

2*x - y[x] + y'[x]*(2*y[x] - x) == 0

The solution says "The functions N(x,y(x)) and M(x,y(x)) can be identified from the differential equation:"

y1=Integrate[(-s+0),{s,0,x}]
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus

Hello Richard,

An update has been made per your feedback. Please delete the related saved file from your cloud storage to see the updated version. https://www.wolframcloud.com/browse#Home/Copied%20Files

Thank you,
Christine Owens
Wolfram U Project Manager
Wolfram U
POSTED BY: Christine Owens
Phil Earnhardt
Phil Earnhardt
Posted 13 days ago

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.
POSTED BY: Phil Earnhardt
POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking

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?
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus

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]
POSTED BY: Richard Daehler-Wilking

It does seem like the symbol f has some other definition attached to it. If you clear f before evaluating the derivative, it should give the expected result. For example:

Clear[f]
D[Exp[f[x]], x]
POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking
POSTED BY: Luke Titus
Phil Earnhardt
Phil Earnhardt
Posted 15 days ago

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?
POSTED BY: Phil Earnhardt

You can get different sized arrows using VectorScaling and VectorSizes. For example:

VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, VectorScaling -> Automatic, VectorSizes -> {0, 1}]

The advantage of using all vectors with a unit length and specifying the magnitudes with colors is to make the plot less cluttered when vector lengths differ a lot in the plot. When using the default unit lengths for the vectors, you can get a legend that shows the length that corresponds to a particular color using PlotLegends

VectorPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, PlotLegends -> Automatic]
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 16 days ago

enter image description here

Can r be i?
POSTED BY: Tingting Zhao

Sure, but most (maybe all?) introductory diff. eq. courses restrict their attention to real-valued solutions. [This is a practical convention, since the course usually follows first-year, real-valued calculus.]
POSTED BY: Michael Rogers

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.

POSTED BY: Luke Titus

Regarding Lesson 3 and the associated Problem 5: At first I was baffled why the instructor did so much work to come up with a solution that he then verified using DSolveValue[...]. Why not simply use DSolveValue[...] to begin with, as was done in the recent course on Partial Differential Equations. Then I realized that this instructor is actually teaching us math, not simply Wolfram Language functions!
SUGGESTION (in terms of Problem 5 of Problem Session 1, but can be applied to the population model in Lesson 3 as well): After the text "Move dx to the right-hand side and use Integrate on the left-hand side", show the result of moving dx to the right-hand side BEFORE integrating:
(1 / {f(x) + 1)) df(x) = 1 * dx
NOW point out that we have separated variables and can integrate each side separately. (The instructor said this, but we didn't have this form of the equation to look at, so I missed his point.)
POSTED BY: Richard Daehler-Wilking

Thank you very much for your comments, Richard. Yes, the main focus of this course is the learn the math and theory behind differential equations while using the Wolfram Language simply as a tool to do some of the work. I appreciate your comment about how we should show more of the intermediate steps and explicitly show what is being integrated before doing the integration. We will take this into account when making future improvements to this course.

POSTED BY: Luke Titus

This is just a reminder that the Differential Equations Study Group begins tomorrow (Monday, April 13).

The Study Group will offer an excellent opportunity to get certified in Differential Equations with guidance from our popular instructor, Luke Titus.

I look forward to seeing you all!
POSTED BY: Devendra Kapadia
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