# Solving Second-Order Linear Differential Equations With Taylor Series

Hello,

This was a Calculus “Challenge Homework Problem” that I did as I take the University of Pennsylvania Calculus course at coursera.org.

I have never learned this method of solving a second-order differential or a first order differential problem. I find it to be very useful. What you see here is only part of the useful process.

If you read the note in the problem statement, the professor, which is a tough professor anyhow, states that this problem is tough.

I can happily say that I solved the problem[1] –green check mark. đź™‚

References:

[1] Harding, Chris. (2018, Feb. 24). Solving Second-Order Linear Differential Equations With Summations. (2018, Feb. 24). scribd[com] Available from: https://www.scribd.com/document/372329980/Solving-Second-Order-Linear-Differential-Equations-With-Summations ; Google URL: https://goo.gl/aZtkW8

# Solution: Tough Vector Calculus Problem. I would need infinite time and resources to solve this problem in my opinion

To me, this is a tough problem. I don’t think I would have solved it, the final solution and answer, with infinite time and resources.

I do provide a solution that I figured out from the hints from the solutions manual. I included the solution below[1].

References:

[1]Â Harding, Chris. (2018, Feb. 11). Solution: Tough Vector Calculus Problem. I would need infinite time and resources to solve this problem in my opinion. Retrieved (2018, Feb. 11). scribd[com]. Available from: https://www.scribd.com/document/371287480/Tough-Vector-Calculus-Problem ; Google URL: https://goo.gl/RKRa84

# Difficult Sophomore Level Chemical Engineering Material Balance Problem: 1000 Gallon Capacity Brine Problem

Hello,

As I refresh my mathematics as a form of psychological mindfulness, I run into problems that remind me of my time in chemical engineering. Below[1] is a problem that would fit nicely in the sophomore level chemical engineering class called material balances.

References:

[1]Â Harding, Chris. (2018, Feb. 10). Difficult Sophomore Level Chemical Engineering Material Balance Problem: 1000 Gallon Capacity Brine Problem. Retrieved (2018, Feb. 10). scribd[com]. Available from: https://www.scribd.com/document/371233104/Moderately-Difficult-Sophomore-Level-Chemical-Engineering-Material-Balance-Problem-1000-Gallon-Capacity-Brine-Problem ; Google URL: https://goo.gl/tgSABi

# First-order Linear Differential Equation and Integrating Factor. A Derivation of Newton’s Law of Cooling

Hello,

I am a medically retired chemical engineer and I am refreshing my mathematics, learning C++ programming, and refreshing my basic engineering as a psychological form of mindfulness.

Every once in a while, I find a problem that I like a little more than others. Since I am in calculus, there are many problems that I like, in truth, but I am a slow typist and I am learning LaTeX.

Anyhow, what engineer would not like one of Sir Newton’s derivations[1]?

References:

[1] Harding, Chris. (2018, Feb. 07). First-order Linear Differential Equation and Integrating Factor. A Derivation of Newton’s Law of Cooling. scribd[com]. Available from: https://www.scribd.com/document/371027275/Calculus-Heat-Transfer-Problem ; Google URL: https://goo.gl/mx6imi

# Numerical Method Reviews

Hello,

This note will be a collection of my personally created numerical method reviews. I will be reviewing topics from two books[B1;B2].I have uploaded my PDFs to scribd because I have had hit and miss experience with LaTeX in WordPress. Also, the PDFs can be accessed, downloaded, and printed easily.

I have chosen books [B2; B5] because they are used in courses at MIT that are freely available online. Since I am using Octave, I found a great book for that environment too[B1]. I plan to study both books.

Purpose

Most books have an initial chapter that discusses pertinent topics to be successful for the rest of the book. The topics are briefly discussed, and the authors [B1;B2] really expect the reader to have full knowledge or gain full knowledge of the material before proceeding.

I am entering a learning phase and a review phase. I am reviewing calculus [B3] and algebra[B4], but I am learning linear algebra[B5]. This note will be a place that I categorize my reviews.

I am writing these reviews with LaTeX, so that takes time as well. Still, LaTeX lets me easily write in the language of mathematics once I learn it well.

Numerical Methods Using MATLAB Reviews[B2]

Quote: It is assumed that the reader is familiar with the notation and subject matter covered in undergraduate calculus sequence. This should have included topics of limits, continuity, differentiation, integration, sequences, and series. Throughout the book we refer to the following results.”[B2]

Quote: “Example 1 is about as easy a limit proof can get; most limit proofs require a little more algebraic and logical ingenuity. (sic: read basic knowledge should be well understood.) The reader who finds “Î´ – Îµ ” discussions hard going should not become discouraged; the concepts and techniques are intrinsically difficult. In fact, a precise understanding of limits evaded the finest mathematical minds for centuries.”[B3]â€ť[B2,1]

Reviews For Numerical Methods Using MATLAB[B2]:

[B2,1] Harding, Chris. (2015, Dec. 21). Review of Limits. Retrieved (2015, Dec. 21). Available from: https://www.scribd.com/doc/293834966/Review-of-Limits

Scientific Computing With MATLAB And Octave[B1]

Quote: â€śIn this book we will systematically use elementary mathematical concepts which the reader should know already, yet he or she might not recall them immediately.

We will therefore use this chapter to refresh them and we will condense notions which are typical of courses in Calculus, Linear Algebra and Geometry, yet rephrasing them in a way that is suitable for use in Scientific Computing.â€ť[B1]

Reviews ForÂ Scientific Computing with MATLAB and Octave[B1]:

[B1,1] Harding, Chris. (2015, Dec. 26). Chapter 1, What Can’t Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices.(Octave) Retrieved (2015, Dec. 26). Scribd[online]. Available from: https://www.scribd.com/doc/294069551/Chapter-1-What-Can-t-Be-Ignored-1-Section-1-1-To-Addition-And-Multiplication-Review-Of-Matrices

Books:

[B1] Quarteroni, Alfio; Saleri, Fausto; Gervasio, Paola. Scientific Computing with MATLAB and Octave. Fourth Edition. Retrieved (2015, Dec. 18). Shannon[online]

[B2] Mathews, John H.; Fink, Kurtis, D. (1999). Numerical Methods Using Matlab. (Third Edition). Prentice Hall

[B3] Anton, Howard. Calculus with Analytic Geometry. Fifth Edition.

[B3a] Barker, William H.; Ward, James E. (1995). The Calculus Companion to Accompany Calculus with Analytic Geometry. Fifth Edition by Howard Anton. John Wiley

[B4] Grossman, Stanley I. (1992) Algebra and Trigonometry, Second Edition. Saunders College

[B5] Strang, Gilbert. (2009). Introduction to Linear Algebra. South Asian Edition. Cambridge University Press India Pvt. Ltd.