Solving Second-Order Linear Differential Equations With Taylor Series


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

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. 🙂


[1] Harding, Chris. (2018, Feb. 24). Solving Second-Order Linear Differential Equations With Summations. (2018, Feb. 24). scribd[com] Available from: ; Google URL:



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].


[1] 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: ; Google URL:


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


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.


[1] 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: ; Google URL:


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


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]?


[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: ; Google URL:


Numerical Method Reviews


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.


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 “δ – ε ” 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]”[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:

Scientific Computing With MATLAB And Octave[B1]

Quote: “In 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.”[B1]

Reviews For 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:


[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.

Limits Review: A Rigorous Approach

Limits Review: A Rigorous Approach


Dr. Anton–Calculus Textbook–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 “δ – ε” 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.”[3]”[1]


I have hopes to learn some Numerical Methods[2] to allow me to contribute to PLOTS ( Numerical methods are  often used by engineers to mathematically solve problems that do not have “exact” solutions. In truth, they can be used to solve problems that have exact solutions where the solution is quite difficult as well.

Also, the studying of math and engineering help me mentally by providing the psychotherapy methods of mindfulness and compensatory cognitive training for my schizoaffective disorder (bipolar type).

Here, I have provided my first LaTeX created PDF on this topic– LaTeX is great for writing mathematics and is free–that reviews limits and absolute values. I am writing such documents because it is likely that I will lose focus or be hospitalized since that has happened in the past. I can then read a document I made to provide an understanding to move forward. In this case, one of my numerical methods Books started with a vague and abstract definition of the rigorous limits approach[1;2], and I wanted a better understanding[1].

I plan to do similar PDFs for other “review” topics in Section 1.1[2]. By the way, I picked[2] because MIT has a free course that uses the same textbook.

In my reference[1], I also provide a great book that covers the use of Octave and MATLAB when doing numerical methods.

Great Video and Special Note

This MIT video on the Rigorous approach has great information of absolute values, etc. In fact, I suggest the reader pay particular attention, within the video[3], of the power of a simple absolute value property of “Triangle inequality”

|a + b| \leq |a| + |b|

that is discussed during the problem solving near video time of 35:12 to end at 46 minutes.

The professor is being kind. To be good, one must have an amazing memory and an amazing grasp of the fundamentals of mathematics. Really, I believe one should obviously begin this grasp in each math class, but, if one is intelligent enough to grasp future fundamentals ahead of the curve, then he or she can go back and review past sections as they pertain to the current set of problems. I am not that person, but hope to gain enough of a “working” knowledge that I can be competent and useful in the world of PLOTS. With that said, all the video[Embedded; 3] is important and a good test of one’s knowledge on this topic[1].



[1] Harding, Chris. (2015, Dec. 21). Review of Limits. Retrieved (2015, Dec. 21). Available from:; Google URL:

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

[3] MIT OpenCourseWare. (2011, May 05). Unit I: Lec 5 | MIT Calculus Revisted: Single Variable Calculus, Unit !: Lecture 5: A More Rigorous Approach to Limits. Retrieved (2015, Dec. 28). youtube[online]. Available from:


An investment opportunity : For those interested in OxiGene (OXGN)–a developer of novel anti-cancer drugs


First, I own OxiGene (OXGN) stock. Also, I have two STEM degrees: chemical engineering and biological sciences, so I have some understanding of the science behind OxiGene drugs. As such, I am not an expert, but I can provide a somewhat educated evaluation.


OxiGene once traded at near $8,000 per share, $5,000 per share, $3,000 per share, and from $1,000 per share to the current, December 26, 2013, near $2.60 per share[6]. It is a small cap company with no current product on the market[3;7]. As such, its stock dropped over the years because of the recession and negative Earnings Per Share, EPS. What should be noted is that small-cap stocks are the usual winners of a recession recovery[4]. Also, the splits make the share price less.

Clinical Phases, Orphan Drugs, and Pipeline

With that said, Zybrestat, its leading drug, was in phase 3 for Anaplastic Thyroid Cancer (ATC)[12] and designated an orphan drug by the United States (US)[14;15], but, according to information from an OxiGene source, Zybrestat will not currently enter continued phase III trials for  ATC in the USA because of funding. As such, I assume OxiGene will not be marketing Zybrestat for ATC in the USA. Still, according to FDA website, OxiGene’s combretastatin drugs have been designated Orphan drugs for “Treatment of anaplastic thyroid cancer, medullary thyroid cancer, and stage IV papillary or follicular thyroid cancer”, “Treatment of ovarian cancer”, and “Treatment of acute myelogenous leukemia.”[16, search: combretastatin]. US Orphan drug status guarantees OxiGene exclusive marketing rights for seven years and various significant tax breaks[15]. According to the US Food and Drug Administration, Phase 3 can take from 1 to 4 years[13]. At the OxiGene website, a graphic indicates that the Zybrestat Phase 3 for ATC is half way completed, but I received a phone call from an OxiGene representative that said phase 3 for ATC will not continue in the USA due to FDA requirements for a large population of patients, near  300, in a future phase III trial. The FDA required number of subjects is, at this time, to costly for OxiGene. The representative did say that OxiGene is still planning to continue USA clinical trials for ovarian cancer. Once Zybrestat is approved by the FDA, assuming phase 3 trials continue to go well[5], a physician, oncologist as an example, can prescribe Zybrestat for other cancers, which is called “off-label” use[2], in ethical situations. In truth, I assume FDA mandates on ATC clinical trials could be an indication of future FDA mandate on ovarian cancer trials. OxiGene is also expecting early marketing of Zybrestat in the European Union, EU. Also, Zybrestat has been designated an EU orphan drug for Anaplastic Thyroid Cancer, ATC, and ovarian cancer. The OxiGene representative that contacted me also said OxiGene is still planning to market Zybrestat in the European Union once approved. Eventually, I have hope the stock will stabilize at a new price significantly greater than where I bought it.

As an example, current and past phase I thru III drug trials of OxiGene drugs, Zybrestat and OXi4503, show promise with Anaplastic Thyroid Cancer (ATC)[12], Ovarian Cancer[12], Acute Myeloid Leukemia (AML)[10;11], myelodysplastic syndrome (MDS)[10;11], and other abnormal vascular diseases such as macular degeneration[12]. Zybrestat has shown success against Pancreatic Neuroendocrine Tumor Model as well[9], and Zybrestat has had mixed results with non-small cell lung cancer[12]. This is important because investment companies look at “how many” people a drug will serve to earn profits per share. Remember, a physician can prescribe for “off-label use,” but the additional prescriptions might or might not affect the overall share price. Finally, Zybrestat has a good safety profile so far[5]. According to reference [5], Zybretsat did show some myocardial ischemia, but the OxiGene representative told me the ischemia problem has been eliminated by prescribing a drug that helps with hypertension. As the reader may know, the US FDA denies the approval of many drugs because of heart issues, but I cannot definitively say this is why the US FDA required near 300 subjects for a Zybrestat phase III trial for anaplastic thyroid cancer, ATC. As mentioned previously, an OxiGene representative told me that OxiGene will not currently be completing ATC phase 3 clinical trials in the USA because of funding.

Research Pipeline

OxIGene also has an effective research pipeline. One of OxiGene’s drugs in the research pipeline is benzosuberenes[1], which have shown success for prostate, ovarian, and non-small cell lung cancers[1] in the pico-molar range. OxiGene owns the rights to all of Baylor Universities studies on benzosuberenes. If you notice, the Division of Cancer Treatment and Diagnosis, National Cancer Institute at Frederick, National Institutes of Health, Frederick, Maryland, United States was apart of the following journal article too[1]. In truth, I do not believe the US Government wants OxiGene to fail. I believe the science behind OxiGene’s drugs is strong[1;3;5;7;8;9;10;11;12]. For a nice video of Zybrestat in action within a biologically active cancer tumor[8].

Low Stock Price

Although OxiGene has performed a reverse stock-split, which indicates a distressed company that is possibly facing NASDAQ delisting if the stock does not increase, I believe the strong science and importance of cancer therapy make the risky company a good buy. Still, I am not a financial analyst. With that said, I would ask a potential financial analyst to strongly consider the science behind OxiGene’s drugs when making a decision.


[1] Tanpure, Rajendra P; George, Clinton S; Sriram, Madhavi; Strecker, Tracy E; Tidmore, Justin K; Hamel, Ernest; Charlton-Sevcik, Amanda K; Chaplin, David J; Trawick, Mary Lynn; Pinney, Kevin G. An Amino-Benzosuberene Analogue That Inhibits Tubulin Assembly and Demonstrates Remarkable Cytotoxicity, Jun 2012. Medchemcomm[online]. 2012. vol. 3(6). pp. 720-724. Available from: PMCID: PMC3682772; NIHMSID: NIHMS386131; doi: 10.1039/C2MD00318J

[2] Off-label Drug Use: What is off-label drug use?[online]. 2013. Available From:

[3] Murphy, Bryan. Yes, This Time Really is Different for OxiGENE (OXGN), Nov 6, 2013.[online]. 2013. Available from:

[4] Consumer Reports Money Advisor. Post-recession investing: No matter which kind of economic recovery we get, diversification still wins, Oct 2009.[online]. 2013. Available from:

[5] Kanthou, Chryso; Tozer, Gillian M. Microtubule depolymerizing vascular disrupting agents: novel therapeutic agents for oncology and other pathologies, Jun 2009. Int J Exp Pathol[online]. 2009. vol. 90(3). pp. 284-294. Available from: PMCID: PMC2697551; doi: 10.1111/j.1365-2613.2009.00651.x

[6] NASDAQ. OxiGene, Inc (OXGN) Interactive Chart Analysis.[online]. 2013. Available from:

[7] Oxigene. Clinical Trails.[online]. 2013. Available from:

[8] OxiGene. How Our Drugs Work. How do ZYBRESTAT (fosbretabulin) and OXI4503 work?[online]. 2013. Available from:

[9] Wall Street Journal. OxiGene Presents Preclinical Data Demonstrating Significant Antitumor Activity of Zybrestat ® in Pancreatic Neuroendocrine Tumor Model, Oct. 21, 2013.[onlne]. 2013. Available from:

[10] Turner, Daniel BS; Gonzalez, Andres BS; Pettiford, Leslie RN; Meacham, Amy MS; Wise, Elizabeth BS; Bosse, Raphael C BS; Chaplin, Dai PhD; Hsu, Jack W. MD; Brown, Randy A. MD; Hiemenz, John W. MD; Norkin, Maxim MD, PhD; Wingard, John R. MD; Cogle, Christopher R. MD. 1463 A Phase I Study Of The Vascular Disrupting Combretastatin, OXi4503, In Patients With Relapsed and Refractory Acute Myeloid Leukemia (AML) and Myelodysplastic Syndromes (MDS), Dec 7, 2013. American Society of Hematology[online]. 2013. Available from:

[11] OxiGene. Pipeline. OXi4503 in Onocology.[online]. 2013. Available from:

[12] Oxigene. Pipeline.[online]. 2013. Available from:

[13] US Food and Drug Administration. Step 3 Clinical Research.[online]. 2013. Available from:

[14] Landes Bioscience. Oxigene breathes life into cancer treatments. Cancer Biology & Therapy[online]. vol 6(8). Available from:

[15] US Food and Drug Administration. Orphan Drugs.[online]. 2013. Available from:

[16] US Food and Drug Administration. Search Orphan Drug Designations and Approvals.[online]. 2013. Available from: