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A classical math problem: differential equations

LucMeekes
23-Emerald I

A classical math problem: differential equations

Here is the Mathcad 11 sheet that solves nth order differential equations with constant coefficients, referenced in the document thread with similar name here: A Classical Math Problem from a Classical Text (Differential Equation).

Note that this, as it is, is NOT usable for:
- Systems of differential equations,

- Differential equations where the coefficients of the derivatives are not constants (when they depend on t).

But in both cases, similar techniques can be used to solve specific systems.

Luc

13 REPLIES 13
RichardJ
19-Tanzanite
(To:LucMeekes)

Nice, but unfortunately it will not work in any version of Mathcad past version 11. The way you are using symbolic math in that first program was never officially supported, and was removed in version 12. Even if that could be fixed in some way, the Mupad symbolic engine can't handle the invlaplace expression

torme
4-Participant
(To:RichardJ)

In my original post in this thread I was using Mathcad Prime 3.0.  My career with Mathcad is long.  I think I started many years ago with Mathcad 3 way before PTC entered the picture.  My math challenges were relatively simple algebra problems, so I didn't get into much calculus.  In my retirement I'm exploring areas where necessary math skills are more demanding. LucMeekes and Richard Jackson bring up topics unfamiliar to me.  The bottom line, however, seems to be that in addition to Mathcad, pencil and paper are still necessary.

LucMeekes
23-Emerald I
(To:torme)

You can have Mathcad do most of the manipulations and some solving, even "abiding by the rules".

Then you still shouldn't need to use pencil and paper:

Prime should be able to do the same, using it's symbolics.

Success!

Luc

Your's solution is wrong. !You have to check numerically , or otherwise.You can't use this initial contitions like y(Pi/8)=1 only zero in brackets like that:

  • y(0)=(any here), y ' (0)=(any here) ,y ' ' (0)=(any here)....... etc


rys.png

LucMeekes
23-Emerald I
(To:jkowalski)

I think that Thomas and the textbook and myself have come to the correct resulting function,

but you're right: It's best to check the result:

Your definition of y(t) does not simplify the given differential equation to 0,

and the boundary conditions are different.

What did you use to get that symbolic result, and what did you use for the numeric result?

Luc

torme
4-Participant
(To:LucMeekes)

I'm not convinced that the original questions have been resolved.  Is there an easier way, and is pencil and paper necessary?  I just finished the course on Signals and Systems at Rice (Discrete Time Signals and Systems, Part 1: Time Domain | edX) and (Discrete Time Signals and Systems, Part 2: Frequency Domain | edX). I passed the time domain part and flunked the frequency domain part.  I made the mistake of trying to do all problems using MATLAB and never resorted to pencil and paper.  It turns out that half the problems were meant to be solved using pencil and paper.  MATLAB cannot index vectors using zero or negative numbers.  I don't know what the situation is with Mathcad, but my attempts to use Mathcad's signals functions indicated that I have a lot to learn.  My bottom line is still - if you are working with numerical methods it is best to have numbers and not a lot of variables.  Many thanks for the interesting comments and elucidations.

sfan
1-Newbie
(To:LucMeekes)

Hi, Luc,

May I ask the reason that you collect the inverse laplace with sin(2t), cos(2t) and exp(t) in this step:

Capture.JPG

Is it because you found the term on the left side can be traced back to the basic laplace transform like laplace(sinat)= a^2/(s^2+a^2)?

Or collect the inverse laplace with sin(2t), cos(2t) and exp(t) is a universal method in this step?

Thank you for your help!

Shawn

lturek
4-Participant
(To:jkowalski)

Jan Kowalski wrote:

Your's solution is wrong. !

Sorry, but it seems that yours is wrong!

You have to check numerically

Agreed on. So check yours and we get y(pi/8)=0.552 and not 1 as it should be!

You can't use this initial contitions like y(Pi/8)=1 only zero in brackets like that:
  • y(0)=(any here), y ' (0)=(any here) ,y ' ' (0)=(any here)....... etc

Yes, we need the values for t=0 when doing Laplace. But it looks like you haven't noticed that Luc replaced y''(0) (which we don't know) by a generic c and solved later, after doing invlaplace, for c to fulfill y(pi/8)=1.

As you don't show how you arrived at your wrong solution we cannot tell you where you failed.

LT

LucMeekes
23-Emerald I
(To:lturek)

Ah, I see that I have to correct. forgot the sqrt(2) before the sine. Correction:

{Leopold, where does your 0.552 come from?}

Luc

lturek
4-Participant
(To:LucMeekes)

{Leopold, where does your 0.552 come from?}

You have again mistyped Jan's "solution". You have your parenthesis wrong.

With the correct expression all conditions (equation, y(0), y'(0)) but y(pi/8) are OK.

Can't post a screenshot as I had not saved the sheet.

But in the meantime Jan has found his error and came up with the same solution as you did.

LT

LucMeekes
23-Emerald I
(To:lturek)

Ah, these parentheses.

OK, Now I see that Jan equated y''(pi/8)=1:

I had a little mistake.Correct one is:

rrr.png

Even his aa.png made ​​a mistake.

sfan
1-Newbie
(To:LucMeekes)

You are a genius!

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