Talk:Systems of Linear Differential Equations

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Diana's feedback from 7/26 on the last 2/3 or so of the page can be found at --Chris 16:21, 10 August 2013 (EDT)-

--Chris 12:04, 25 July 2013 (EDT) Xintong, I made a number of small edits myself; you can view them in the History if you'd like. Below are my edits of the first part of the page until the math got the better of me. I've asked Diana to finish editing the rest of the page.

Basic Description:

  • Intro to Systems of Linear Differential Equations
    • differential equation: Consider changing "involves" to "contains".
    • linked system "multiple different"? This seems redundant or awkward wording.
    • Last two sentences. How does eigentheory connect with the last sentence?

More Mathematical Explanation:

  • Solving the Two Dimensional System
    • Intro sentence needed, something like: "In order to understand how a system of linear differential equations is useful in making sense of the Cold War Arms Race, we first need to understand the general system. ¶ The general system of linear differential equations is..."
    • Do we want to write "the coefficients" before "a,b,c, and d are elements of ALL…"?
    • Remove the sentence beginning with "Naturally, the systems…"
    • P5: Change "intuition" to "understanding".
  • Back to the Two Dimensional System
    • P1,S1: Help the reader understand why the solution to the one dimensional case suggests that particular solution. It's not obvious at least to this reader.
    • P1,S3: You introduce λ and vector v here without any explanation.

Checklist for Writing Pages

Original Checklist completed by --Xtian1 16:26, 23 July 2013 (EDT)

Messages to the Future

  • I felt a bit rushed at the end, so I think I only glazed over the concepts in the Stability section. The "Why it's Interesting" section feels a bit forced out too. If someone ever wants to enhance this page further, feel free to flesh out these sections.
  • Someone who has real analysis background: Please look at my More Rigorous Proof and add more to it, such as proving facts about eAt.

References and Footnotes

  • All images are cited.
  • I don't see any places that I really need references. Someone let me know if they find something.

Good Writing


  • Explains how linear differential equations are very applicable to the arms race. The Cold war arms race was used to generate interest in the reader.

Quality of prose and page structuring

  • Beginning part of the page mentions how the page is about systems of linear differential equations.
  • Each section is definitely relevant.
  • Section headings help clarify section's content.
  • Links into many Helper pages for maximum clarity.
  • The hardest mathematical part, the More Rigorous Proof, is placed at the end of the More Mathematical section.

Integration of Images and Text

  • Each image is vital in helping understand the material.
  • When relevant, text points to the most relevant part of the image.

Connections to Other Mathematical Topics

  • Abundant use of other Math Images pages.

Examples, Calculations, Applications, Proofs

  • Many examples are provided.
  • Proved most of the things, though left a few advanced proofs for others.
  • Many applications, such as arms race and Hooke's law.

Mathematical Accuracy and Precision of Language

  • I tried to be as mathematical as possible without losing clarity.
  • To my knowledge, everything I wrote is mathematically accurate.
  • Defined many foreign terms to help the reader's understanding.


  • Less images than desired, but this is a very equation-based topic.
  • Paragraphs are typically short, with equations spaced out nicely.
  • Hided various points to decrease "scariness".
  • Minimized white space.
  • None of my text will wrap around images, since my images have to be big to be clear. Computers with different resolution screens could screw it up, so I decided to not bother with text wrapping.
  • All images are in the correct location.
  • No weird computer code in hidden sections.
  • Tried on a few computers, seems all fine.

Older Feedback

General Feedback

My detailed feedback from 7/12/13 can be viewed here (much of this has been addressed since the 12th). The major points from that document are in the appropriate sections below.

-Diana 11:59, 15 July 2013 (EDT)

Basic Description

Regarding ways to make the main image more interesting – as we mentioned, don’t be afraid to sort of sensationalize the image. One option would be a full vector field representation of the solution to your example overlayed on a photo of an atomic bomb detonation…

At the end, explain more clearly what a, b, c, d represent. As a verbal statement of this system, you could add a few sentences right under the equations saying that, to represent the negative and positive relations, the US lowers its budget by a dollars for every dollar it spent the previous year, and raises it by b dollars for every dollar increase in the USSR’s budget… etc. Also, mention that x’(t) is the derivative of x over t, and remind the naïve reader that this represents how much the US spending changes over t years.

-Diana 11:59, 15 July 2013 (EDT)

--Chris 10:35, 1 July 2013 (EDT)::*I like very much that you chose the Cold War topic for your beginning example as a way of engaging the reader's attention.

As you write the Basic Description, imagine that your audience is a student just beginning the study of Calculus; this person understands that Calculus deals with the nature of change but needs a step-by-step walk through of equations, using both text and graphics as support for making sense of how Calculus can be used to help understand the Cold War.

--Gene July 10; * "the two powers inevitably fought a political and psychological war for the second half of the 20th century"; don't you mean DURING "the second half of the 20th century"? That's not what the war was FOR.

--Chris 11:50, 12 July 2013 (EDT) * This section is a major improvement over what you had previously. I'll provide more detailed feedback later.

Introduction to Systems of Linear Differential Equations

Make the second bullet point the last bullet point; your current 3rd and 4th bullets are more closely related to the first. Also, “linear combinations” is not necessarily a term everyone will know – it should have a balloon or a link or a following explanation/definition.

-Diana 11:59, 15 July 2013 (EDT)

More Mathematical Section

Solving the Two Dimensional System

--Chris 11:50, 12 July 2013 (EDT) Peng will serve as point person for review and help on this section.

I notice that you followed up on feedback from Wednesday and showed how you to get the one-dimensional equivalent of "u" and made a subsection solving for du/dt = ku.

The One Dimensional Case

Back to the Two Dimensional System

I know the idea of "guessing" the solution was bothering you, and this could seem weird to the reader, too. Think of going back to your one-dimentional expression for the general solution, u = Cekt. A good bet would be that the solution to our 2-D case is similarly, u = vert for some vector v and constant r. That would mean u' = rvert. But we also know u' = Au. So rvert = Au = Avert. Then (rI - A)vert = 0. Dividing by the (definitionally non-zero) ert, we have rv - Av = 0, or rv = Av, the expression solved by eigenvector v and eigenvalue r.

-Diana 11:59, 15 July 2013 (EDT)


We discussed linking this section back to your Cold War scenario. Just phrase your scalars and initial conditions in terms of what they would mean in that scenario, and connect the solutions to the Cold War idea by explaining their implications in that context.

-Diana 11:59, 15 July 2013 (EDT)

The More Rigorous Proof

Regarding the idea that A0 = I, this is generally considered more a definition of A0 than a property to be proven. But we can understand it as a result of maintaining consistency within the rules of matrix multiplication: I = A-1A1 = A0. Of course, we can’t always rely on this, because the property A0 = I holds for non-invertible matrices, as well. But for the purposes of this page, we can assume our matrices are always invertible, since a non-invertible matrix implies at least one zero eigenvector, which would simply make one row of our matrix trivial – it could be removed to make all eigenvectors non-zero.

During and at the end of the proof, as we discussed, linking to the 2-dimensional case and to the Cold War idea would be helpful and more original here. The “what does it mean?” That you’re launching into after this is also very valuable on that front.

-Diana 11:59, 15 July 2013 (EDT)