Here is the link to the Differential Equations series: https://www.bigmarker.com/series/daily-study-group-intro-to-differential-equations/series_details

ok, the first list is slope, got it,

Function does not evaluate its second argument automatically, so you could use Evaluate as shown below to achieve the desired result in the first case:

In[1]:= sol = DSolveValue[p''[x] + 2 p'[x] - 3 p[x] == 0, p[x], x]

Out[1]= E^(-3 x) C[1] + E^x C[2]

In[2]:= solution = sol /. {C[1] -> 1, C[2] -> 0}

Out[2]= E^(-3 x)

In[3]:= verify = 
 p''[x] + 2 p'[x] - 3 p[x] == 
   0 /. {p -> Function[{x}, Evaluate[solution]]}

Out[3]= True

A link to the download folder containing all the lecture notebooks (as well as any additional materials used during the sessions) is included in the reminder email you receive from BigMarker. It will also be shared in the live session everyday of the study group series.

Thank you. I wonder which is a better way to avoid a naming conflict: localizing p[x] using , say, Module, or by using

/. {p -> Function[{x}, Evaluate[solution]]}

Could you please provide an alternative example? Thanks. M

Thanks, Vincent. Could you please explain what's the meaning of "1" in the list. Is it the module (length) of the vector which is, in fact, variable (but this is apparently handled by color coding) or something else? The lecture said that the "1" is D[x,x] , but why are we taking this derivative?

Thank you.

Hello Michael,

In Function[x, body], the variable 'x' is the first 'argument' and the expression 'body' is the second 'argument'.

The variable 'x' is effectively a 'local' (Module) variable inside Function and hence, as shown below, the function evaluation does not depend upon its name.

IIn[1]:= f = Function[x, x^2];

In[2]:= g = Function[a, a^2];

In[3]:= f[5]

Out[3]= 25

In[4]:= g[5]

Out[4]= 25

In[5]:= f[x]

Out[5]= x^2

In[6]:= g[x]

Out[6]= x^2

Also, technically, Function has the attribute HoldAll:

In[7]:= Attributes[Function]

Out[7]= {HoldAll, Protected}

In practice, this means that the second argument 'body' is not evaluated by Function, so Evaluate is required below (since 'body' is defined outside Function).

In[8]:= body = x^2;

In[9]:= Function[x, body][5] (*incorrect*)

Out[9]= x^2

In[10]:= Function[x, Evaluate[body]][5] (*correct*)

Out[10]= 25

Hope this helps.

If there is just one argument the { } can be omitted. For multiple arguments

f = Function[{x, y}, x*y];
f[3, 4]
(* 12 *)

Is it fair to say that DSolveValue [...] is simply equivalent to DSolve [...] [[1,1,2]]?

In the Lesson 7,

the first equation 5 e^y(t)+5 t e^y(t) dy/dt =0 is equivalent to dy/dt = - 1/t; which is trivially solved in one line: y(t) = - ln(t) + c What is the purpose of multiplying both sides by "5 e^y(t)" and making the simple solution more complicate? Is it done for purely *didactic * reasons?

Friday sessions for this series will start at 10:30 AM CT with the review session for the week running from 10:30-11:00 and the day's lesson beginning at 11:00 AM CT.

Hello Zbigniew,

The second one fails because Derivative requires a Function input as shown in In[3] below.

In[1]:= solution = 90 + C1 Exp[1/3 t];

In[2]:= (p'[t] == 1/3*p[t] - 30) /. p -> solution

Out[2]= Derivative[1][(90 + C1 E^(t/3))][t] == -30 + 1/3 (90 + C1 E^(t/3))[t]

In[3]:= f'[t] // InputForm

Out[3]//InputForm=
Derivative[1][f][t]

The problem is that you have assignments (=) instead of equatity (==) for y[0] and y'[0]. This should work:

DSolveValue[{4*y''[t] + 6*y'[t] + 2*y[t] == 0, y[0] == 5, y'[0] == 1/2}, y[t], t] // Expand

Note, however, that after you have evaluated the wrong DSolveValue expression, then Mathematica has saved the two assignments (y[0] = 5 and y'[0] = 1/2) and will interfere when evaluating the corrected expression.

One way to fix this is to restart the kernel after fixing the expression (e.g. using the Quit[] command).

Thank you, Devendra. Very nice: With "All", it displays the second element of all the solutions/assignments. M

PS It seems to me that in the early versions there was a spot reserved for the arrow "->" , so that {a -> 3.}[[1,2]] would return "->". Am I mistaken?

Exercise 10(4)

y1[x_] := x^(1/2 (1 - Sqrt[5]));
y2[x_] := x^(1/2 (1 + Sqrt[5]));

What are the advantages of using [29]-[30] instead of [38]-[39]?

n[29]:= (x^2 y''[x]-y[x]==0)/.**y->Function[x,y1[x]]**//Simplify

Out[29]=True
In[30]:= (x^2 y''[x]-y[x]==0)/.**y->Function[x,y2[x]]**//Simplify

Out[30]=True

In[38]:= (x^2 y''[x]-y[x]==0)/.**y->y1**//Simplify

Out[38]=True

In[39]:= (x^2 y''[x]-y[x]==0)/.**y->y2**//Simplify

Out[39]=True

And similarly with the exercise 9(3) Thanks

Sorry, the original expression in the lesson is: [Phi](x,g(x)) = g(x)ln(x)+3x^2-2g(x).

Hi Michael. Yes, while the exponential term dominates the long term behavior, the factor of x should still be kept. I will fix that in the notebook for that exercise.

Hi Michael. That term is there mainly to demonstrate the concept of an exact equation and the methods to solve them. You could definitely cancel that term and solve the equation much easier, but it wouldn't demonstrate how exact equations can be solved. Since you can cancel that term to get a simpler equation, it is a nice way to check the result obtained when using the methods of exact equations.

OK. I suspected that this was a didactic tool. Thanks for confirmation.

Exercise 13(1). Just for practice, I tried to do it without "manual labor" while still reproducing all the steps of defining the "undetermined coefficients" . However, the implementation is sloppy. For instance , using "DeleteCases" to remove a bunch of 0s produced on the previous step, seems redundant. Apparently, I did not find the right version of the "CoefficientList[]" Please, advise.

eq =y''[x] + 2 y'[x] + y[x] - x^2 - 3 Sin[x] == 0;

parSol[x_]:= a Sin[x] + b Cos[x] + c x^2 + d x + e;

collConst = Collect[eq[[1]] /. y -> Function[x, parSol[x]], {x, Sin[x], Cos[x]}]

Out[39]= 2 c+2 d+e+(4 c+d) x+(-1+c) x^2+2 a Cos[x]+(-3-2 b) Sin[x]

In[40]:=
Solve[DeleteCases[Flatten[CoefficientList[collConst, {x, Sin[x], Cos[x]}]], 0] == Table[0, {5}], {a, b, c, d, e}]

Out[40]= {{ a->0, b->-(3/2), c->1, d->-4, e->6 }}

Both ways will get the job done. However, using Function is a little more versatile as it can handled pretty much any form of the function. For example, if you had defined y1[x] instead as y1 = x^(1/2 (1 - Sqrt[5])), then y -> y1 wouldn't work and you would need to use y -> Function[x, Evaluate[y1]].

Thank you. Still, it does not look pretty. It would be good to tune up the action of CoefficientList, so that it considers only non-zero elements. Am I asking too much ? :)

PS: It can be done by using If[], but it would not look neat either.

https://www.bigmarker.com/series/daily-study-group-intro-to-differential-equations/series_details

Sorry for asking, but where should we look for the Quizzes? Is it at the usual "amoeba site" or anywhere else? Thank you.

From what you said, I take that using the "Function" is a more secure, bullet-proof approach. It is good to know! Thank you.

In lesson 11, slide 12, the solution of the problem in Example 1 is defined by means of the ASSIGNMENT, "=", whereas in slide 14, the analogous solution of the problem in Example 2 is defined by means of the DELAYED ASSIGNMENT, ": =". Traditionally, functions in Mathematica are defined via the delayed assignment. Is there a reason for the cited difference, or it simply doesn't matter? Please advise!

Attachments:

Delayed_OR_NOT.jpeg

No, I didn't do it. But, an excellent hypothesis, Hakan! It prompted me to investigate this matter further. I originally downloaded the materials before the meetings started. Then additional materials were added, so after one week into the course, I downloaded all the materials again (and I keep both zip files). It is there where this change took place. I downloaded the materials again a few minutes ago, and the "error" is there--you can confirm it yourself.

My question stands: is it an error or a correction? Or, perhaps, it doesn't matter, as both versions seem to work equally well.

Just above the output cell bracket you will see a tiny little cell bracket. Select that, then (on Windows) go to the menu Cell -> Cell Properties then select "Open"

Thank you, Hakan for confirming my observation.

Wow! I would have never guessed it. So far I have been closing the code cells by double-clicking their output cell bracket. I've been a Mathematica user for a long time, but I didn't know that there was another option to do the same by selecting the code cell and then going to Cell >> Cell Properties and clicking or unclicking the box next to "Open" (in Linux). Again, I find it very interesting and, I bet, so would many other users. If I may, I suggest you point out this option during the Friday review. Many thanks for this pointer!

Is there a Level 1 Certification available for this study group?

[WSG21] Daily Study Group: Differential Equations (begins November 29)

How to copy a Quiz in the local NoteBook?

I second that , The best WolframU course so far ......

OK, that's why I really need to get an account ! (?) Now I will follow your and Arben's advice and write Wolfram-U a letter :)

PS Just did it.

Section 5, Exercise 2 The radius of convergence can be calculated using SumConvergence:

In[2]:= SumConvergence[n (2 x + 1)^n, n, Assumptions -> x [Element] Reals]

Out[2]= -1 < x < 0

Therefore, the series converges whenever -1<= x <= 0, so the radius of convergence is 1/2.

In the previous statement the original inequality -1 < x < 0 is substituted by -1<= x <= 0*. However, they are not equivalent, because the Sum [n,{n,0,Infinity }] does not converge.

To avoid the confusion, the "=" sign should be dropped.

• The situation is slightly different in Exercise 1. Addressing this difference would be helpful for revealing a nuance in the definition of the radius of convergence.

Even though I'm registered for this study group, I stopped receiving notifications about the upcoming meetings as well as the follow-ups about their recordings. I would appreciate receiving those announcements.

Hi Michael, Oh, I have missed this one. Thanks for the pointer. Best, Zbigniew

When clicking on the Feedback button in the online version of the course, it gives an "HTTP Error Code 404 - Oops, the page you're looking for can't be found.".

Before that is fixed, should I report comments / questions on the online version via the usual Wolfram support channels (i.e. email)?

Again, I would like to know how equation (1), in lesson 20, Special Functions, on slide 12 (see also my previous note), was derived.

Below is my attempt at it, in which I ended up with an expression involving the EllipticF function.

Wow! Surprisingly, it turns out that this expression is actually equivalent to the one involving the hypergeometric function. By the way, I'm amazed by the symbolic power of Mathematica in finding the integral and even more in finding that the two expressions are equal to each other. If I were to do it by paper and pencil, it would be ...

But the question still remains: how did you come up with (1)?

Attachments:

pick2.jpeg

That was just to simplify the discussion. The method works when they are general analytic functions, but the term analytic wasn't defined at that point.

Hi Zbigniew. I did not derive that equation, but you can find a reference here: https://aip.scitation.org/doi/10.1063/1.4922268

Hi Hakan, Please email wolfram-u@wolfram.com directly about any feedback related to the course.

Our College math professor called HGF "The mother of all bombs". And he really meant it, being in his youth a part of the "Russian Manhattan Project". I should also mention Dr. Eric Weinstein and his MathWorld (a part of WR) - the outstanding Encyclopedia of Math, which (in conjunction with Wolfrfam Demo projects) allows to test many concepts using the Mathematica.

PS: Morse and Feshbach, in my view, is still a great resource to be kept in the vicinity of your desk :)

Same here

Thanks, Luke, I really appreciate the pointer. Cheers, Zbigniew

Thank you, Zbigniew. It would be a pleasure to meet in person. However, I have an additional suggestion. As we are "here and now", and already found a common denominator, why don't we try doing together something useful for our learning of Differential Equations and Mathematica, and may be even beyond. I was thinking of writing a "computational essay" on the topic dealing with the non-linear oscillations (and, naturally, with the Differential Equations). And your last question sounded to me like a natural part of this project. The problem that I have in mind deals with a simple "(almost) High school level" system, which may display, (almost) in spirit of NKS, very complex behavior. A while ago, we have done some ground work with my son, and I can send a short note on the subject (for anyone interested in this endeavor).

PS This is not just a personal message. Anyone on the forum, who is interested, is welcome to join this discussion (some background in Math and physics would be very helpful). I am very grateful to the Wolfram-U team and personally to Luke for creating such an inspiring and productive atmosphere.

Please email wolfram-u@wolfram.com

Michael,

Awesome idea, and I'm game. Definitely! I'll be mostly free after I give and grade the final exam in my undergraduate Fluid Mechanics, i.e., probably in two days. I would love to get involved in such a project. I'm not sure how long this discussion board will be available. So for this kind of collaboration, I propose that we deal with each other directly via email and/or skype or zoom. My email address is below.

Best wishes, Zbigniew

Giles and West House Faculty in Residence

919-667-5150 (c)

Zbigniew J. Kabala, PhD, PE, Associate Professor

Duke University

Department of Civil & Environmental Engineering

E-mail: kabala@duke.edu

http://www.cee.duke.edu/faculty/zbigniew-j-kabala