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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
55 Replies
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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

In Problem Set 4, Problem 7 says to solve an initial value problem with this differential equation:

y''[x] - y[x] == Exp[2]*x^2   (* Note the constant exponent 2 for Exp[] *)

However, the solution of the problem implicitly assumes g[x] equals Exp[x]*x^2 , where Exp[ ] has an argument of x instead of 2. I had no trouble solving the problem as stated with Exp[2], but perhaps that's a typo in the problem?
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
POSTED BY: Richard Daehler-Wilking

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

I"m trying to work Problem 5 in Problem Session 4:

Find a general solution for the following equation using Variation of Parameters: y''(x)-6y'(x)+9y(x)=sin(x)

Compare with the particular solution obtained by the Method of Undetermined Coefficients. ==> The solution goes into great detail presenting the method of undetermined coefficients. It then finishes with "It can be observed that this is equal to the particular solution derived from Variation of Parameters." However, I can't see where variation of parameters was used.

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

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}}
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

Hello Richard,

Thank you for your feedback. I've created a ticket per your report for Luke and our team to review. As soon as I have more information, I'll let you know.

Best,
Christine Owens
Wolfram U Project Manager
Wolfram U
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

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.

POSTED BY: Luke Titus

In the Basic Example of Lesson 13 (Method of Undetermined Coefficients) , DSolveValue returns a solution in terms of Cos[x] and Sin[x]. QUESTION: Is it possible to specify which basis functions DSolveValue uses? That is, can we ask DSolveValue to return an answer in terms of

Exp[x] and Exp[-x]?
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

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

Thank you for pointing that out. The Integrate code shouldn't be there. In place of the Integrate code should be 

m = 2 x - y;
n = 2 y - x;

We'll get that fixed. Thanks again for pointing that out.

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

Lambda is chosen just by convention. It's the symbol a lot of differential equation textbooks use. There isn't a meaning behind why Lambda is chosen, it's just the symbol that has commonly been chosen in the past so we decided to use the same symbol just to be consistent with other references.

POSTED BY: Luke Titus

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.
POSTED BY: Richard Daehler-Wilking

Here is a link to the course: https://www.wolfram.com/wolfram-u/courses/mathematics/introduction-to-differential-equations/
POSTED BY: Luke Titus

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.
POSTED BY: Richard Daehler-Wilking

I see. You should be getting an email after each lesson with a link to the recording. I'll talk to Cassidy today to see if we can resend those links to you.

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

Someone from the WolframU team will reach out to you shortly.

POSTED BY: Luke Titus
Phil Earnhardt
Phil Earnhardt
Posted 12 days ago

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. :)
POSTED BY: Phil Earnhardt

THANK YOU. I really appreciate your taking the trouble to address my difficulty. MY REAL PROBLEM may simply be impatience. I just now tried the e-mailed link again, and this time the screen says "The webinar will begin momentarily ..." Apparently it doesn't work if you try it too soon.
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

Basically, you need to convert everything to a string then use <> to combine the strings you want. You can use quotes to convert the C[1] expressions to a string, but you need to use the ToString function to convert the iterator in the Table to a string. For example:

In[94]:= Table["C[1] = " <> ToString[i], {i, -2, 2}]
Out[94]= {"C[1] = -2", "C[1] = -1", "C[1] = 0", "C[1] = 1", "C[1] = 2"}
POSTED BY: Luke Titus
POSTED BY: Richard Daehler-Wilking
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
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
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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