Sunday, June 30, 2013

More Linear, Homogeneous Systems of Differential Equations

Once more we return to homogeneous linear systems of differential equations, to see how the solutions can be broken down and what we need to do. This time we will look at this example :

dx1/ dt = x1 + x2

dx2/dt = 4x1 + x2

  Note the first step, again, is to form a matrix from the coefficients, which we see are (1, 1) for the top, and (4, 1) for the bottom. Thus:   A =

(1 .....1)


Then, it must be true from the properties of determinants that:

(A - l) D =

[(1 - l).......1] [k1]

[4 ..... .(1 -l)] [k2]

Again, we allow l  to be subtracted from the first element in the upper left, and from the last element in the lower right). Cross-multiplying and using matrix properties we obtain the characteristic equation:

l2 - 2l - 3 = 0

where l1 = 3 and l2 = -1 We need to first find a vector that solves the equation: (A - l1 I)D = 0
In the first instance, we substitute the first eigenvalue, l= 3, into the matrix, whence:

(A - 3I) D = 0 =

[-2.....1] [k1]

[4 ...-2] [k2]


-2k1k2 = 0, and  4k1 – 2k2 = 0,  So k1 = ½ and k2 = 1

Then our first eigenvector is: K1 =



Therefore, the first linearly independent solution for the system is:

X1 = K1 exp (l1 t) = K1 exp (3t) 

The second eigenvalue was l2 = -1 so we repeat the process again to obtain the equation to be solved:

(A - l2 I)D = (A - (-1)I) D = (A + I)D

Then, (A + I) D = 0 =

[2.....1] [k1]

[4 ….2] [k2]

Or: 2k1 + k2 = 0 and  4 k1 + 2k2 = 0

So that: k1 = 1, then k2 = - 2

The second eigenvector is then: K2 =



So another linearly independent solution is:

X2 = K2 exp (-t)

Then add the two solutions, to obtain: 

X = X1 + X2 = K1 exp (3t) + K2 exp (-t)

With, of course, the 2 column vectors (as computed above) substituted in for K1, K2. This serves as a general approach for solving all such systems.

Problems for the Math Maven:

1) Obtain the general solution(s) for the system:

dx/dt = 2x + y

 dy/dt = 2x + 3y

2) Obtain the general solution(s) for the system:

dx1/ dt = 3x1 - 4x2

dx2/dt = x1 - x2

Horace Cooper: Another Moron Exposed on Bill Maher's 'REAL TIME'!

Voter Fraud ‘Study’ Authored By Republican Who Pleaded Guilty In Abramoff Scandal

We need to really thank Bill Maher, on his 'Real 'Time show, for highlighting the dolted Morons for the rest of us. Such was the case Friday night, as Horace Cooper - Conservo commentator and goofball, was exposed as belonging to this elite group. I will say that I only pegged Cooper as a semi-Moron until he opened his yap about climate change. (As Maher pointed out, he sat on the wrong side of every other issue under discussion, so why not this too?)

Cooper interjected at one point, believe it or not: that: "true science must always challenge the status quo or accepted version". He then went on to claim that at one time the "flat Earth" was the true accepted "science" and the spherical Earthers had to challenge it! Huh?    In other words, this dolt was comparing the currently accepted science of global warming (based on thousands of papers published in peer-reviewed journals)  to the flat Earth view!

In fact, as far back as 240 B.C. it was known by the much more enlightened Greek astronomers, in particular Eratosthenes, that the Earth was spherical, not flat. This was opposed to the primitive flat Earth perceptions (NOT science!)  embraced by the sheep herder tribes of Judea, many of whom scribbled down their Bronze Age myths in the Bible. In Eratosthenes’ case, around 240 B.C., he had to first decide what exactly he had to measure to assess sphericity as opposed to circularity. This is where a key assumption entered: that the Earth was spherical and the Sun distant enough that its rays at Earth were essentially parallel.


Eratosthenes thereby performed a measurement of the angle of elevation of the Sun at noon at Alexandria, and at Syene. This value could then be used to obtain Earth’s circumference, at 250,000 stadia or about 24,900 miles. The repeat of the experiment from thousands of different directions, orientations, shows sphericity not just circularity. Why? Because if Eratasothenes (or any of his thousands or millions of followers – who repeat the experiment even today at assorted universities) were measuring a circle, they’d have to be on the circle’s EDGE to obtain its circumference . Cut out a circle from cardboard and examine it. Any distance on the circle itself would be a chord, not a circumference. One would have to stand or situate exactly ON THE EDGE to get the circumference. No untrained, non-mathematical ancient mind would remotely contemplate this, because to him one would “fall off” at the edge. (Gravitational physics would need another 2500 years to be developed by Galileo and Newton)

Thus, the very act of measuring a circumference using a shadow angle (and trigonometry) on any part or place of Earth implicitly presumes its sphericity , since the extension of all such measuring lines leads to a circle that can be oriented around any direction across Earth. What then is the sphere? Technically – as we see from calculus, it is the integration of an infinite number of conic sections that are each circles – which results in a sphere (see diagram below).

How did the ancient Greek astronomers (e.g. Eratosthenes) break out of this and arrive at sphericity? In Eratosthenes’ case, around 240 B.C., he had to first decide what exactly he had to measure to assess sphericity as opposed to circularity. This is where a key assumption entered: that the Earth was spherical and the Sun distant enough that its rays at Earth were essentially parallel.

Recall here that roundness (circularity) and sphericity are two different properties. One (circularity) applies to a simple two dimensional surface or geometry. Indeed, the property of circles was investigated by Sumerian and Egyptian mathematicians long before any of the biblical authors emerged from their caves. This is why one biblical reference often cited by literalists (Isaiah 40:22: "“It is he that sitteth upon the circle of the earth..." ) cannot be assumed to mean the ancient Judean sheep herders knew anything about the Earth's sphericity. No, they only possessed a crude 2-dimensional perception: i.e. of themselves situated at the center of a circular FLAT disk with edges.

Thus, mathematically to advance, we take a circle – say defined by:

x2 + y2 = 4

And rotate it around the x-axis to generate a sphere. This is done by using calculus to integrate: p(4 - x2)dx from (-2) to +2. Thus, in the process of rotation an infinite series of circles is generated, to obtain a sphere with radius 2 and volume 33.427 cubic units. That the integration yields cubic units – proves that the result is a sphere since circles lack volume as geometrically defined.

Of course, one could never expect a numbnut like Cooper to know any of this, but then he ought not have compared the flat Earth paradigm to current climate science, which is actually ass-backwards. It is the current denial paradigm which is evocative of the flat Earth in relation to global warming-  which is the true analog to the spherical Earth.

In one sentence, then, Cooper exposed himself as a full moron whose bloviations on anything to do with climate change can be ignored, just as an earlier Maher guest ('Kennedy' ) claimed warming was due to 'Sunspots!" ( see e.g. )  Meanwhile, co-panelist Dan Neil of the WSJ at least got in one of the last words when he pointedly noted that he could see the effects of global warming-climate change by merely observing the manifestations outside, in the environs he inhabits. Which makes one wonder what peculiar environs Cooper inhabits that he can't see them.

Thanks, Bill, for every now and then clearly showing whose voices we need to ignore on which subjects! (As we've since learned to ignore Niall Ferguson's on "no evidence of water contamination"  from fracking!)

Saturday, June 29, 2013

Thoughts and Lessons from a Brother’s Death

Image may contain: 2 peopleJ
, John with Janice when we visited him in Las Vegas in 2009

Earlier this month my second brother John died, after nearly a month being in a vegetative state and connected to life support. He’d suffered a massive stroke during the 2nd week of May, which I didn’t even learn about – how could I?- until returning to the U.S. from Europe. I immediately began to beat up on myself, because I’d promised to phone him again before leaving to plead with him to get the cardiac open heart surgery he needed to save his life – but which he rejected nearly a month earlier. Doctors had warned him it was perhaps the only chance he’d have for life, given the extent of arterial blockage,  but he didn’t want to take any chance on going under the knife.

Obviously, in any serious medical procedure, there exists a chance something might go amiss, and you may not come out of it. A simple colonoscopy can result in a perforation of the colon wall and death. A prostate biopsy, such as I had last July, can result in hemorrhage or sepsis. A prostate cancer treatment – whether surgery or radiation- carries its own severe risks. But we get these things done – on weighing the probabilities, and possible negative against positives – because we rationally view them as essential to continue life or quality of life. Alas, John did not, though my middle brother Jerome and myself pleaded with him to change his perceptions.

We also pleaded with him to at least stop smoking, which the doctors had warned him about, but he was unable to do so. Having already gone blind nearly a year and a half earlier (from diabetic retinopathy) he wasn’t about to give up any more sensual stimuii. Given the loss of visual input, he compensated by audio –Ipod ……and non-stop smoking as a prime activity. One might say that between the refusal to get the cardiac operation and the inability to stop smoking….at least so much smoking….he essentially laid a path that was ultimately unfortunate for him.

What lessons to take away from John, his life and especially latter decisions, actions?

- It is by no means as easy to change a person as you might think.

I know, I know, this is almost a truism but is no less germane for being so. While I regret not having known Johnny better, mainly on account of spending a good chunk of years outside the U.S., I do recall on the few intermittent occasions I did see him, or have contact, trying to entice him to make better life choices. I either appealed to the range of alternatives he seemingly couldn’t see (such as going to Vermont in 1998, as opposed to going to Las Vegas where he fell into a gambling addiction) or making better health choices to extend and enhance his life, life quality – such as cutting down on the cokes and smokes, eating a better diet… opposed to going to Vegas buffets. But in the end, he always made his own choices, and they usually weren’t in his favor or best long term interests.

-It is possible John had a methylated personality which prevented him from making the optimum choices over his life(see e.g. )

As noted therein, according to Epigenetic theory (see, e.g. DISCOVER, May, pp. 50-55)any traumatic experiences – even in the genetic past- are capable of leaving molecular “scars” adhering to our DNA. While the DNA itself is unchanged, behavioral alterations are imposed by the addition of the chemical interloper. More to the point, these alterations can be genetically passed on. In our family’s case, just tracing assorted genealogical lines near and far has disclosed numerous episodes of such trauma, as well as earlier relatives who likely succumbed to it. They also made terrible life choices and died or committed suicide at even younger ages than John’s. Earlier relatives were scarred by the First World War, and earlier ones by the Black Death ravaging Germany from 1634-1660. (Many of our more distant ancestors in the Brumbaugh line lived in the Bayerische region we visited, which included a Catholic church (St. Sebastian's) in Partenkirchen with a cemetery where over 1,100 were buried in a mass grave- all victims of plague.)

Could a methyl group then have hitch hiked on John’s DNA making him more likely to make poor choices? This is very plausible. But in the end we can’t know for certain because no actual chemical tests have been done.

- In the end, we need to appreciate and try to help our loved ones as much as we can while they’re alive and not postpone until later showing our care and affection

  As I wrote on John’s Memorial page: “This loss of John ought to bring all the remaining sibs closer together, realizing life is too short.” Family squabbles, what sib was favored over who by parents, who is a “real Christian” and who isn’t, or why that atheist sib needs to marginalized, and who is Dem and who is Repub- --- all these fall into insignificance when one considers the sheer brevity of human life. Thus, when we sow or fuel conflict we are basically wasting time – the already brief time in this life- on literal minor beliefs, ephemeral differences and subjective viewpoints.

At least so far, John’s sibs have come closer together, and one hopes it endures, and doesn't fade with time or memory loss.

John, wherever you are- if indeed in a ‘heaven” as proposed by our family's believers- or existing as nonlocal quantum wave forms (e.g. )

Be assured we will try to live up to the Self-potential you yourself sometimes glimpsed for your own life, but which bad luck,  poor decisions.....or lack of ample love and support when needed....prevented you from achieving!

Friday, June 28, 2013

Mail Call Brane - Readers Seeking Answers

Could you please complete the solution to the ship problem differential equation in your blog of May 12? I know you started the solution for many of us but I can't seem to finish it! -
Andy K., Omaha NE

Let's pick it up where the partial solution left off.  I.e. you had to integrate:

dv/ (g - av) = dt

To obtain:

ln (g - av) = -at + C

The hint was then provided that you had to obtain the value of  a  to get the integrating factor.

As given in the partial soln.:

(k/m) = a

=   (10, 000 f/s lbs.)/ (4 x 10 6 sl) =  2.5 x 10 -3

Now, taking natural logs of both sides of the integrated equation and simplifying::

  v =  g / a    -   B (exp - a t)

But the condition from rest is v= 0 at t = 0,  Þ   B = g / a

So:  v = g / a   - [1 - (exp - a t)]

=  g / a  - [1 - (exp - 0.0025)t]

Then, if  g / a  = 20 f/s, v = 20 [1 - (exp - 0.0025)t]

And from this the terminal velocity can be found.  Remember as v approaches the terminal velocity the second term must approach 0. so effectively the terminal velocity is: v = 20[ 1 - 0] = 20 f/s

Which equals 13.63 mph when converted from  feet/ sec  to miles per hour.

Q. I must have missed something! In your June 4 blog post you explained how the Germans you met at Zugspitze aren't buying Tom Hanks' version of events in what you describe as an up and coming "HBO 13-part series". Could you elaborate a bit because I'm sure I must have missed some back story! - Jake, Oklahoma City, OK

A. This stems from the news last year that Hanks told in an interview that he intends to "do the American public a service" - because he thinks they "have been snookered into believing that Lee Harvey Oswald was framed."  So, he plans to do a 13-part HBO series to remedy that situation - or so he believes. And he intends to use the mammoth book 'Reclaiming History" by Vince Bugliosi as his "historical" basis. Of course, the book is nothing of the sort, and has numerous problems of fact, history and logic which I already expatiated on at some length, e.g.





Clearly, Hanks' hubris has gone to his head, along with his 3 Oscars and some success in making HBO docu-histories, such as The Pacific. I mean, cripes, the guy actually believes he's chief historian of the country despite the fact he probably can't even pass a basic test on the assassination! See e.g.

Readers interested in other takes on Vince Bugliosi’s debauchery of history can check out these links:

Q. I had been on a high T regimen to build more muscle mass, but then last year started to develop pains in my lower back. It got so bad I had to see a doc, and strangely enough, he wanted a PSA test done. Well, I had it done and the result came back at 7.3. He now wants me to get a biopsy but after reading your account of it, and then the treatment options for prostate cancer I'm not sure I want any part of it! I know you will probably say to get it done, including treatments, but do I have any other options here? - Tony M., Sacramento, CA

A. Well, you could try watchful waiting. You don't give your age but that may only be a solution if you're over 65 or so. If you're a young man then I'd say you've got zero options left and you need to get the biopsy. It's the only way to ascertain if you have prostate cancer, where it's located and how aggressive it is.  Readers can see my account of the biopsy I had here:

The problem with waiting if the cancer (usually an adenocarcinoma) is aggressive, it that you may run out of options. At least at the early stage, say several cores and no Gleason scores over 3 +3 or 3+4 you have a range of treatment choices. But once the threshold is crossed to Gleason scores 4+3 or 4+4 or higher, you're basically down to the most extreme response which is generally a radical prostatectomy, and often adminstration of androgen suppression hormones afterwards.

My suggestion? Find a urologist or have your physician refer you to one, and get that biopsy! Oh, and leave those gels of T alone!

Q. I want to get your take on some recent responses to Edward Snowden's travels, especially this line that we've heard from Jay Carney, John Kerry and others in the administration:

"Mr Snowden's claim that he is focused on supporting transparency, freedom of the press and protection of individual rights and democracy is belied by the protectors he has potentially chosen: China, Russia, Cuba, Venezuela and Ecuador,"

Margaret S., Cincinnati, OH

A. What you referenced is the usual sort of false analogical rot trotted out by paid lackeys and other servants of the State and other State-System-Big Media actors. Can you take it to the bank? Hardly!  First, Snowden's ultimate and most important objective was preserving his own freedom, as opposed to wasting away in an Army Brig somewhere, being left naked for 24 hrs. a day for weeks (as Bradley Manning was), subjected to intermittent sleep deprivation and fed bread and water (ok, I exaggerate a bit on the last but only a bit). Hence, the choice of where he went must be seen in that light, i.e. with primary goal of avoiding the U.S. 'Gulag 'at all costs - after likely being given justice at the hands of a kangaroo court "grand jury" assembled out of Alexandria, VA (the domain of CIA, NSA, FBI)...

Also, to ensure no intervention he had to select nations that at least had some kind of power or leverage to stand up to a raging Uncle Sam. China and Russia met the criteria, for differing reasons.

The portrayal of both China and Russia as specious destinations for Snowden is absurd: China is now our leading trade partner for Pete's sakes - also our leading lender for all those Social Security, DoD and other monies! Russia assisted us with the apprehension of the Tsarnaev brothers, real terrorists!  Also, keep in mind Hong Kong wanted him out asap. He was a  PR liability and it was only on account of large pro-Snowden protests from thousands of people (and maybe some slight arm twisting from the mainland)  they laid off him, and let him leave HK for Moscow. Let's also bear in mind that China is now the largest capitalist nation in the world, and basically our banker! So, if we are calling down opprobrium on that country,  WTF are we doing taking out loans for military weapons from them etc.? Talk about hypocrisy (but of course, that's our nation's biggest export) - see also:

As for Russia, Putin also isn't exactly enamored of having Snowden holed up there at Moscow’s Sheremetyevo Airport. Note also: the airport is not on Russian soil, so technically Putin isn't obliged to extradite him, or indeed, pay any attention to a huffing and puffing bunch of Amerikkan hypocrites who seem to forget or ignore their own lawlessness the past 12 years, including starting illegal wars violating Article IV of the Nuremberg laws, see also:, not to mention violating the Geneva conventions with rendition and torture (water boarding).

As for Ecuador, I have news for the Amerikkan bigots and hyper-false patriots: Most of Latin America and the Caribbean sees us as a Yanqui gangster state after the  CIA assisted some reactionary Venezuelans (including Luis Posada Carriles, a former CIA agent)to  blow up a Cubana Airlines passenger plane over Barbados on Oct. 6, 1976.

One thing to take away from all of this: the US of A is not as pure as the driven snow as it claims or portrays itself. It also has supported terrorists, as in Honduras and Guatemala in the 80s, and also helped bring bin Laden to fruition by leveraging him against the Russkies when they invaded Afghanistan n 1979. But see, then it was convenient and expedient, to advance U.S. geopolitical objectives. Never mind the geopolitical strategies, a terrorist can be a state actor and hide behind all kinds of excuses, rationales.

Imho, the worst calamity to happen to the world was the collapse of the bi-polar equilibrium that prevented one country from gaining too much hegemony and power in the world. That happened with the collapse of the Soviet Union. Given that one big "superpower" now tries to rule everyone, it's no surprise a lot of smaller players- and even big ones-  would like to jab a finger or two in its hypocrtical "eyes."

Most interesting to me has been who exactly has come out on each side. It's now become much easier to tell the pro-state purveyors of deceit and character assassination from real patriots invested in protecting liberties. Ed Schulz, for example, is now dismissed by me as a bloviating propagandist having called Snowden a "punk" and demanded his return and prosecution. Chris Hayes, meanwhile, has been pro-Snowden - if not as outspoken as some others. But his intelligent take gets kudos from me. Rachel Maddow? I've no idea where she sits because she's presented some segments on her show but hasn't disclosed her position. Nice and coy, and safe!

The media corporo-pundits  like Aaron Ross Sorkin and David Gregory? Blow dried hacks, both of whom ought to be sent to Komodo Island on one way tickets - mainly to provide protein for the giant reptiles there. Oh, and you can toss Dianne Feinstein into that stew too, along with John 'Wuss' Kerry and Dick-head Cheney!

Thursday, June 27, 2013

Solution of Linear, Homogeneous Differential Equation Problem

Note the first step, again, is to form a determinant (matrix) from the coefficients, which we see are (2, 3) for the top, and (2, 1) for the bottom. Thus:  A =

(2 .....3)

Then, it must be true from the properties of determinants that: (A - lI) D =

[(2 - l)......3] [k1]

[2 .......(1 -l)]  [k2]

Note how we allow l ('lambda') to be subtracted from the first element in the upper left, and from the last element in the lower right). Cross-multiplying and using matrix properties we obtain the characteristic equation:

(l2 -3l +2) – 6 = 0  or    l2 -3l -4 = 0

So that:  (l - 4) (l + 1) = 0,    where l1 = -1 and l2 = 4

We need to find a vector that solves the equation: (A -  lI) D = 0

In the first instance, we substitute the first eigenvalue, l1= -1, into the matrix, whence:

(A - (-1))D = 0 =

[3.....3] [k1]

[2  ...2] [k2]

3k1 + 3 k2 = 0,  and 2k1 + 2 k2 = 0

So: k1 =1  and    k2   = -1

The first eigenvector is then: K1 =



So: X1 =  K1 exp (-t)

The second eigenvalue was l2 = 4 so we repeat the process again to obtain the equation to be solved:

(A - l2 I)D = (A - (4)I) D = (A + I)D = 0 =

[-2.....3] [k1]

[2 ...-.3] [k2]

Or: -2k1 + 3k2 = 0 and 2 k1  - 3k2 = 0

So that: k1 = 3  and k2 = 2

The second eigenvector is then: K2 =


  And another linearly independent solution is:  X2 =  K2 exp (4t)

Because of the principle of linear superposition, one can also add the two solutions, to obtain:

X = X1 + X2 = K1 exp (-t) + K2 exp (4t)

Will Congress Double the Debt of College Students on July 1st? Probably!

Student Loan
 Disaster Warning
When I paid off all $4k of my college loan debt, one year after leaving Peace Corps in August, 1975, it was a day to truly celebrate. More to the point, I'd done it by rigorously saving while in Peace Corps (even on a $125/month stipend) and also after landing my first job after the Corps - working as Head of the Science Dept. at a Barbadian Secondary School.

But make no mistake, that feeling of debt liberation never would have been remotely feasible had I not had the benefit of National Defense student loans at 1 percent interest! This was made possible under the National Defense Education Act (NDEA) of 1958, and the fact my first degree was in a hard science subject qualifying under the Act. The release of loan debt at such at early phase, then made it possible for me to marry in 1975 (the month I left Peace Corps) and also launch out on an independent life with my new wife.

Pity the poor college grads today, however, who are saddled with atrocious aggregate loan debt that now exceeds $1 trillion, and is set to get much worse. The Huffington Post in March, referencing a  New York Federal Reserve report, pointed out that the proliferation of indebted students and their families had knock -on effects on multiple other areas of the economy, including:

- Fewer people with student loans are buying homes

- Fewer people with student loans are able to secure and maintain decent credit ratings

- Delay in buying a car, hence having lower access to job availability

- Postponement of marriage

- Inability to rent because of high debt to income ratio, so having to live with parents

- Too many graduates (and even non-grads) encumbered by student loans can't pay them off efficiently because they can't access the quality jobs to do so (half of all the grads are found in jobs that don't require a college degree). They may then be prey to loan sharks or having liens placed on accounts, property.

As Malcolm Harris has long pointed out, since most student loan debt is government- backed  and can never be discharged, the type of meltdown the student loan explosion could precipitate is taking a different shape than the mortgage crisis, with the victims (student borrowers) already in place and struggling.

And the tragedy is that it's set to get worse, much worse.  On July 1, 2013 interest rates on many new federal student loans are set to DOUBLE (from 3.4% to 6.8%)  – meaning any college  education gets more expensive and more out of reach for millions of students and middle class families. All the problems highlighted above will therefore become much more intractable.

Readers should see some notes of resonance from what transpired last summer, when rates as on the Stafford loans, were also due to jump. But students took to Facebook and Twitter to barrage their reps and the higher debt noose was postponed. A patchwork law was implemented to keep loan rates the same, but that expires on Monday.

Basically, absent further congressional action, students are screwed as the interest rate on new subsidized student loans is scheduled to go up again. To stop this from happening, President Obama put forward a long-term solution that cuts rates this year on nearly all new loans, ensures that all students have access to affordable repayment options, and does not charge students higher interest rate to pay for deficit reduction. Sadly, the Repuke congress doesn't appear to be taking him on, and indeed, given their austerity mindset, is fully prepared to let students suck salt matter how many FB messages and Tweets they dispatch.

On the (slightly) positive side, the impending July 1st  rate hike will affect only undergraduates with subsidized loans, where the federal government absorbs some of the interest rate. This makes up about a third of government student loans, and they are awarded based on economic need.  Other undergraduate and graduate students who take out unsubsidized loans, have already been paying interest rates set at 6.8% since 2007.   But the point is that many more will now be put at the higher interest-debt level and that's not news students of modest means and their families want to hear.

As the website put it:

"Now is not the time to make school more expensive for our young people. As our economy continues to recover – at a time when interest rates are at historic lows -- the 7 million students who rely on these loans to finance their education shouldn’t face higher debt as they work to graduate, start a career or buy a house."

I could not agree more!

Wednesday, June 26, 2013

Back to Differential Equations!

Recall that the most straightforward type of differential equation to solve is a linear, homogeneous equation.

Definition: The differential equation dy/dx = f(x, y) is said to be homogeneous if there exists a function g of one variable such that: 

f(x, y) = g (y/x)

Example: dy/dx = ln x - ln y + (x + y)/ (x - y)

satisfies the definition. Why?

Well, because we can recast it in the form:

dy/dx = ln (x/y) + (1 + y/x) / (1 - y/x)

so that:  g(w) = - ln w + (1 + w)/ (1 - w)

meeting the formal condition of the definition.  The point is that any homogeneous equation can be reduced to a separable equation.

One can also solve systems of linear, homogeneous equations. To define homogeneous in this more complex situation, consider the system as shown below:

dx1/dt = a11(t)x1 + a12 (t)x2 + ..a1n(t)xn + f1(t)
dx2/dt = a21 (t)x1 + a22 (t)x2 + ..a2n (t)xn + f2(t)
dxn/dt = an1 (t)x1 + an2 (t)x2 + ..ann(t) xn + fn(t)

where the coefficients aij and the fwhere the coefficients aij and the fi are functions continuous on a common interval, I.

When fi(t) = 0, and i = 0, 1, 2.......n, this system is said to be homogeneous. Otherwise, it is called non-homogeneous.

The key to solving such a system is that one can use matrix form to do so (recall past blogs on matrices, e.g. and in particular something called "the fundamental matrix".

If then, X1, X2....X3 is a fundamental set of solutions of the homogenous system:

X' = AX

on an interval, I, then its general solution on the interval is:

X = c1X1 + c2 X2 + .......+ c2 Xn

= c1[xi1j + c2 [xi2j + ...... cn [xinj]

where the brackets enfold row vectors

Now, let:

X1 = [xi1j] and X2 =[xi2j] and Xn = [xinj]

be a fundamental set of n solution vectors of the homogeneous system X' = AX on an interval I.

Then the matrix:

M(t) =




is said to be a fundamental matrix of the system on the interval.

We want to proceed by solving the linear, homogeneous system:

dx/dt = 2x + 3y

dy/dt = 2x + y

The steps to be followed are:

1) Find the eigenvalues from the matrix composed of coefficients

2) Having obtained the eigenvalues, obtain the first set of eigenvectors (EV1) as shown

3) Obtain the corresponding equation (in k1, k2) for the first set of (coefficient) "eigenvectors".

4) Thereby obtain the second eigenvalues and eigenvectors and the corresponding equations.

5) Find  the primary eigenvectors K1 and K2 . (K1 and K2 are formed from k1 and k2)

6) Write the general solutions (X1, X2) in terms of K1, K2.

7) Thence, write the general solutions of the system, viz. X = c1 X1 + c2 X2


Use the information provided in the graphic frame shown to complete the general solution.  Hint: You should find in a succeeding step that K2 =