Back to Differential Equations

First, let's look at the odd problem solutions:

1)Show that y = cx^2 - x is a solution of the DE: xy' = 2y + x

We first take:

dy/dx = y’ = 2cx - 1

Then substitute for y and y’ into the DE:

xy' = 2y + x

-> x (2cx – 1) = 2 (cx^2 – x) + x

2cx^2 – x = 2cx^2 -2x + x

And:

2cx^2 – x = 2cx^2 - x
Hence y = cx^2 – x is a solution.


3) For the differential equation: dy/dx = -x/4ysketch the curve which passes through the point (1,1)

We re-arrange to obtain: -4y dy = xdx

Integrate to get: -2y^2 = x^2/2

Or: x^2/2 + 2y^2 = c

From this one can put in a set of different values of c to generate the family of curves appropriate to the equation.

That is shown in Fig. 1.

On inspection we see one curve goes through the point (1,1) and that is associated with the equation:

x^2/2 + 2y^2 = 2

Or:

x^2 + 4y^2 = 4


5) Find the particular solution for the equation:exp(x) sec(y)dx + (1 + e^x) sec (y) tan(y) dy = 0 (y = 60 deg when x = 3 )

We can simplify and re-arrange the equation to give:

exp(x) dx/ (1 + exp(x)) = - tan (y) dy

then integrate both sides :

ln(1 + e^x) + ln sec(y) = c

and :

c = (1 + e^x)/ cos (y)

Since : (1 + e^x) = c cos(y)

The particular solution is found :

(1 + e^x) = 2(1 + e^3) cos (y)


Substitute the values for x, y :

c = (1 + e^3)/ cos(60) = (1 + e^3)/ ½

c = 2 (1 + e^3)

Let’s now return to differential equations and recap from the perspective of what we’ve already seen:

Given a differential equation F(x,y, y’, y’’…..y^n)= 0

1) A solution containing n arbitrary constants is a “general solution” (the number of such constants equals the number of the highest derivative)
2) A solution obtained from specific values of the constant(s) is called a “particular solution”
3) A solution to the DE which is not obtainable from the general solution is called a singular solution.

From these definitions (especially 1) one can always “work backward” and obtain the original differential equation from a general solution with arbitrary constants c1, c2, etc. known as a “primitive”

For example, given a primitive:

y = c1e^x + c2 e^-x + 3x

find the original DE.

Since we know the general solution we can differentiate twice to obtain the actual DE;

Differentiate once to get:

y’= c1e^x – c2e^-x + 3

and again:

y’’= c1e^x + c2e^-x

Now take:

y’’- y = c1e^x + c2e^-x - (c1e^x + c2 e^-x + 3x)

so: y’’- y = 3x

or d^2x/dx^2 – y = 3x

is the differential equation.

This also shows the (general) basis to solve differential equations of higher order, by successive differentiations. This leads us now to discriminate between the order and the degree of a DE:

The order of a differential equation is the same as the highest ordered derivative in the equation. Thus, in the example above, the order is 2 (since we have y’’)

The degree of a differential equation is the algebraic degree of the highest ordered derivative. Thus, the example above shows an equation of the 2nd order and first degree.

Given the preceding, let's now move on to solving initial value problems - which also make use of the general solution, but now with some inital values imposed.

Example: Given that the general solution of:

dy/dx + y = 2x exp(-x) can be written:

y = (x^2 + c) exp(-x)

solve the inital value problem for y(0) = 2

This means, if y = (x^2 + c) exp(-x)

Now: x = 0 when y = 2, so:

2 = (0 + c) exp(0) = c

Then: y = (x^2 + 2) exp(-x)

then: y' = -(x^2 + c)exp(-x) + 2xexp(-x)



Example (2) : dy/dx = x^2 sin(y)

and y(-1) = -2


As before, we first re-arrange to obtain:

dy/ sin(y) = x^2 dx

which can be integrated (both sides) to yield:

ln(tan (y/2)) = x^3/ 3 + c

or ln(tan(y/2)) - x^3/3 = c

Now, x = -1 when y = -2 so:

ln(tan(-1)) - (-1)^3/3 = c

or:

ln (-pi/2) + 1/3 = c

or:

0.443 + pi(i) + 1/3 = c

so the solution is: ln(tan (y/2)) = x^3/ 3 + 0.443 + pi(i) + 1/3

or: ln(tan(y/2)) - x^3/3 = 0.776 + pi(i)

Implicit differentiation often also finds its way into solving simple first order differential equations.

Consider, for example: dy/dx = -1/2y

One way is to re-arrange to get: 2y dy = -dx

and integrate both sides to get: y^2 = - x + c

Or: y^2 + x = c

On the other hand one may also use implicit differentiation (which will come in handy for more complex problems):

2y(x) y'(x) = -1

and integrate:

2y(x) y'(x)dx -> y^2(x) + x = c -> y^2 + x = c

Example: Show that x^3 + 3xy^2 = 1 is an implicit solution of the differential equation:
(2xy)dy/dx + x^2 + y^2 = 0

on the interval 0 less than x less than 1
We see why that interval is given by sketching the curve (Fig. 3). It is only well-behaved (continuous) between x slightly larger than 0 and slightly less than 1.

We begin by differentiating implictly to obtain:

3x^2 + 3y^2 + 6xy (dy/dx) = 0

divide through by 3: x^2 + y^2 + 2xy(dy/dx) = 0

Then:

2xy(dy/dx) + x^2 + y^2 = x^2 + y^2 + 2xy(dy/dx)


A few problems:

(1) Find the differential equation which has:

y = c1e^2x + c2e^-3x + sinx as the general solution

(2) State the degree and order of the differential equation:

xy(1 + y^2) dx - (1 + x^2) dy = 0

and find the general solution.

(3) Show that 5x^2y^2 - 2x^3 y^2 = 1 is an implicit solution of the differential equation:

x (dy/dx) + y = x^3 y^3

over the interval (0, 5/2)

(also sketch the graph)

Solutions next time!

Insights into Mithraism

Left: Rendering of Mithras, ca. 100 B.C.


In this blog we examine the god-Man myth of Mithras (Mithra.)
In the oldest extant version of the legend of Mithras, found in the the Izeds (28 in number) appearing in the Zendavesta, Mithras was born on December 25th of the Virgin, Anahita.

The notion of being "born of a solid rock" can probably traced to certain writings of Church Father Justin Martyr who analogized Peter as the mystic "Rock" of the Church ("Thou art Peter, and upon this Rock I will build my Church") to the emergence of Mithraism with Mithras. J.M. Robertson ('The Origin Of Mithraic Mysteries: Cosmology and Salvation in the Ancient World') speculates that Martyr's ruminations are "in touch with the Graecized but probably Semitic myth of the Rock-born Agdestis, son of Jupiter". In other words, not peculiar to Mithraism proper, but a subsequent syncretic addition which then became distorted into a "rock birth" by early Christians. Not suprisingly, probably because of the Agdestis- connection, Mithras is derived from the Latin "rock born" and that concept is perpetuated with the name.

A much older version, probably more accurate is cited in Robertson's Pagan Christs (p. 119) wherein he notes that according to a (Vendedadic) legend "Mithras shoots an arrow at a rock, and water gushes forth where the arrow strikes". The passage is believed by some scholars of pagan mystery religions to refer to either an incident where Mithras saves his followers who are lost in the desert, or that he uses his arrow to reveal an analog to the Judaeo-Christian Decalogue.

ADDENDUM NOTE:

The Zendavesta, literally "text and comment," is the doctrine of Zoroaster (Zarathrustra), comprised in eight parts, written at different periods, but of which the earliest have been assigned to the date of B.C. 1200-1000. In its present form it was collected by Ardeshir, the founder of the Sassanian dynasty, from oral tradition, at the time when he re-established the ancient religion of Persia.

Aspects of Mithraism that later filtered into the Graeco-Roman world (in the form of emergent Mithraic cults) undoubtedly came from this text (the Izeds, wherein “Mithras” was chief and ruler) as well as others likely destroyed with the burning of the library at Alexandria. (See, e.g. J.M. Robertson’s ‘Pagan Christs’, 1966) . Many scholars believe the greater details including Mithras’ disciples, the full recounting of his miracles: including walking on water, raising the dead, changing water to wine etc. were in those destroyed manuscripts.

According to Plutarch, Mithraism first arrived in the Roman Empire through the Cilician pirates whom Pompey subdued, then brought back to Rome as slaves. (Many were actually believed to have been soldiers for whom Mithraism would have been the religion naturally embraced).

Mithraism was a mystery religion, so its historical legends would have always been more accessible than its precepts, rituals were held in strictest confidence – available only to the select few - as a secret society would. (Much like today’s FreeMasons who maintain absolute silence as regards their rituals.) As F. Zindler notes, no specific “works of Mithraism are known since like other Mystery Cults, it centered around a secret or body of secrets only known to those initiated into the rites. Most of our knowledge of it is derived from the iconography of its temples”.

A number of ancient authors (e.g. Porphyry) bemoaned the catastrophic destruction of many of the original works at the hands of the Christians. As we know from the example of Mikey, they can’t tolerate other books sitting on shelves next to their own – they either have to burn them or depict them in toilets.

Nonetheless, we do have enough in Ized 28 of the Zendavesta to document much of Mithras' mythological basis. Perusing through it there is no mention of battling a "bull" - which likely originated from the Graeco-Roman version of the myth, which was probably distorted by re-translations over centuries (much the same way the Christian New Testament has been).

In the actual Izeds (#23-#28) there’s no record of any “Bull”. There is a peripheral reference (by Porphyry) to Mithras as "the Bull of Demiurgos and lord of generation". But again, no "battle" with such. (so it's likely corrupted over time). Robertson (p. 121) speculates Porphyry's reference is actually based on a conceptual error mixing up the Mithraic seat at the equinoctial table (assigned to the Ram, Aries) "which also bears the sword of the Ram." He adds (ibid.): "The sword of the Ram is simply figured as a cross, since the sword IS a cross".

Nor is there account of any "battle with the Sun" (though Roman Mithraists, who worshipped in a Mithraeum, may well have introduced this myth synthetically, as a kind of backhand way of vengeance at the Roman Sun deity, Sol Invictus. After all the earliest Roman Mithraists were Cilician slaves brought to Rome under duress.)

We only know that he emerged with the virgin Anahita giving birth, then began his ministry which included numerous sermons, miracles – especially curing lepers and the handicapped. As scholar J.M. Robertson has pointed out: "The doctrine of the Logos - the Incarnate Word- which Christianity absorbed via the Platonic Jews of Alexandria, was already present in Mithraism" (See: 'Pagan Christs’, page 117)


Re: Mithras' birth date on Dec. 25th, one should note that just as Jesus' date of Birth (putatively on the 25th December) was ahistorical, most likely Mithras' was as well. The assignment of ancient god-man births to this date is understandable given the ancients investment in agriculture, and the Winter solstice marked the day of shortest daylight giving way to increasing lengths of day through Spring. (Perfect for planting in the northern hemisphere).

Some who complain about Mithraism insist there's no evidence that Mithras was a great teacher.
But this is belied in Ized 28. (We were introduced to this in our Comparative Religion class at Loyola, in 1966). In brief passages we see Mithras teaching the multitudes about son-ship and stewardship and the Logos-Trinity. In many ways his message echoes the ones seen in the Gnostic Gospels, and adhered to by the Gnostics. He taught that there was one reality and all were part of it. Each man was a divine “son” of the One God, as well as the doctrine of immortality of the soul. He taught that people must see their brothers as they see themselves and treat them the same. He also echoed the ancient ethical code from the time of Hammurabi:

“Do unto others as you would have them do unto you.”

Incredibly, this has found its way into Christian lore and writings, and claimed to be an original statement of Christ.

Another complaint is the parallel between Mithras and Christ breaks down because there's no mention of "12 disciples" This is possible, but the truth is we may never know whether there were actually 12 disciples of Mithras, or 24, 10 or 5 or none. As noted earlier, most of the (Vendidad) and much older Mithraic manuscripts were destroyed when the Christians burned the Library at Alexandria. In Carl Sagan’s COSMOS, Episode 13, ‘Who speaks for Earth?’ he recounts the killing of Hypatia, one of Alexandria’s foremost woman scientists and also the library’s burning. J.M. Robertson ('The Origin Of Mithraic Mysteries: Cosmology and Salvation in the Ancient World', Oxford Univ. Press, 1989) suggests than more than 300 manuscripts documenting the role of Mithras’ disciples were likely destroyed around the same time as Hypatia’s martyrdom.

The best advice here for Mithras naysayers?

“Evidence of absence is not absence of evidence.”


Some other Christians have complained Mithras underwent no bodily resurrection, but was "taken up in a chariot." Again, false! Ized 28 is most exhaustive on this issue and is confirmed in James Pullen’s (Origin of All Religious Myths- The New Testament Exposed) . Mithras was condemned to death, crucified, buried and then rose in an Ascension. No chariots carried him anywhere, so this is likely a derivation from a later Greco-Roman myth (especially the latter, wherein chariots were prominent and also symbolic).

Robertson observes (op. cit.)

"The celebration of the Mithraic mysteries, of the burial and resurrection of the Lord, the Mediator and Savior, the sacrament of bread and water...all these were in practice long before the first Christian Gospel of a Lord who was buried in a rock tomb."

Another common canard is that Mithraists copied from Christians to fabricate their god -Man not the other way around. This is totally false, and can be ascertained from Tertullian’s writings combined with just minimal reasoning ability. My class notes, e.g. from my Comparative Theology class are fortunately still in one of my old file boxes with Exegesis notes. The main theme, to summarize, is that the course was based on the original writings of Church fathers, including Tertullian, Justin Martyr and Julius Firmicus – all referencing Mithraic scriptures and references therein.

After describing the “Lord’s Supper’ for example, Justin Martyr evinces outrage as he writes (from my notes):

“WHICH THE WICKED DEVILS HAVE IMITATED IN THE MYSTERIES OF MITHRA, COMMANDING THE SAME THING TO BE DONE”


Tertullian also bears this out, scolding:

“The Devil, by the mysteries of his idols, IMITATES EVEN THE MAIN PARTS OF THE DIVINE MYSTERIES. HE ALSO BAPTIZES HIS FOLLOWERS IN WATER THAT PURIFIES THEM OF THEIR SINS”

As the notes indicate, a careless Christian reader might jump to the conclusion that the Mithraists copied from the Christians, but this simply isn’t so, on closer examination. The refutation of this notion inheres in the language of the same trio of Christian Fathers. They all speak, for example, of resemblance of Mithraic rites to Christian rites as being “the imitation of devils”.

But why such an obscure use of language, when they could simply have stated the Mithraists IMITATED their own works, and done? Why the need to introduce devils? This presumes a supernatural influence over and above ordinary human copying.

An answer becomes manifest when we examine more closely the words of Justin Martyr, who expressly argued that "the demons ANTICIPATED the Christian mysteries and prepared PARODIES of them BEFOREHAND"! (Pagan Christs, p. 118)

As Justin further writes (cf. from Loyola Theology notes):

“When I hear that Mithra was begotten of a virgin, I understand that the deceiving serpent also counterfeited this.”

Addendum in notes: No one pretends the Pagan virgin myth in general is LATER than Christianity. Justin’s own words: "ANTICIPATED the Christian mysteries" can mean only one thing: he KNEW the Mithraic version PRECEDED the Christian version. What he said also underscored the arguments from Tertullian and Firmicus. He was so incensed that the Mithraists trumped the Christians' Savior myth on the timeline, that he attributed this to demonic work, aka the "imitation of devils".

Robertson (op. cit., p. 121) also cites other common symbols that the Christians appropriated from Mithraic traditions, including the Agnus Dei, or "Lamb of God". As he notes:

"The Christian assimilation of Mithraism is still more clearly seen in the familiar Christian symbol in which Christ is represented as a lamb, carrying by one forefoot a cross."

What about the way in which Christians have consistently copied images of Mithras and used them for Jesus? Robertson again (op. cit.) supplies an answer (p. 124):

"The Christian imitation, took place, be it observed, because the features imitated were found by experience to be religiously attractive."


It’s clear from this that the Mithras myth sticks in the craw of many biblical literalists and even orthodox non-literalists, since their dependency on the notion of one unique "Savior" exposes their egocentric certainty that the Dalai Lama warned about (previous blog). They simply can’t tolerate the idea that their God-man myths were prefigured in earlier pagan God men. While Catholics more or less accept it and shrug it off, since they make less of a big deal about salvation by “believing on the Lord JC”, the evangelicals lose it, since it means losing their unique claim to a pass into their heavenly reward.

What they ought to be doing is THANKING the Mithraists for providing a ready to use template for Christian copyists and re-translators (intent on using their good book in a PR campaign to lure unsuspecting unbelievers) to build their "Savior" myth on, and have everyone convinced he's unique. And......the sole, sure pass into the righteous afterlife (with harps, clouds and all).

Most amazing, despite this possessive protection of their “Savior” they still haven’t been able to explain why an “act of God” destroyed the 62 foot tall Jesus statue near Monroe, Ohio almost a week ago. One bolt of lightning frying the whole figure to a crisp, except for the steel frame. Maybe they’d be more energized had it been a statue of Mithras!

Why Can't All Religious Believers Be Like This?

As I watched a recent interview with the Dalai Lama I marveled at the man's poise, grace and soft-spokenness. In the course of nearly 40 minutes I heard not one vituperative word or any afterlife -salvation threats spoken in rancor and self-righteousness. He spoke briefly of the Buddhist's Nirvana but didn't ordain that everyone in the world must comply....or else.

In other words, they leave people to other paths, without self-serving condemnation or opprobrium. It was certainly refreshing to see, and contrasted with the loathsome tele-evangelists always getting red in the face as they invoke their hell fire on unbelievers, and "the unrepentant".

In his book, Many Ways to Nirvana, the Dalai Lama consolidates his views and puts them into a perspective that is most welcome in a world poisoned by "us against them" thinking and religious hatreds.

One of the most enduring points he makes is on pages 142 and 143 and concerns the development of the human ego. In Buddhist teachings this is suppressed, because all Buddhists understand the dangers of letting it get out of hand, becoming too self-centered, and then negative.

As he noted (p. 143):

"The negative ego is the extreme, self-centered "I". That ego leads to the harm and exploitation of others"

Of course, we know this dynamic well in terms of confidence artists, and their ilk - people like Bernie Madoff who bilked billions from people who trusted them. But these extreme egoists also appear in religious guises - where they're most dangerous. They are, indeed, the "wolves in sheep's clothing" that Christ warned about.

How would one recognize such a wolf? A key clue is found in the Dalai Lama's elaboration of the negative ego entity (p. 161):

"The root of all disturbing negative minds is grasping at things as truly existent".

In the next 12 or so pages, he elaborates what this means. What I will do is try to paraphrase it all in the next eight or nine paragraphs. Since his language is also in terms of the several Buddhist schools (Shravakayana, Pratyeka-buddha-yana, and Mahayana) I will enlist the help of Buddhist philospher Alan Watts (This Is It) as well to make sense of what the Dalai Lama is saying.

The key word is "grasping". Our (human) mind does not see or perceive reality, and because it also GRASPS, it is "difficult to control". Why?

Because the human mind, though short on reality and factual existences, grasps to reach for certainty. But certainty is the very poison that kills the mind. It destroys it from the inside out, by making its possessor believe he has the absolute access to the truth, the whole truth, and nothing but the truth.

This can come in numerous diverse forms, especially in religious faiths or settings, since these attempt to placate followers by offering nuggets of "truth" and abiding verities. Most often, despite there not being one scintilla of hard evidence for an afterlife, religions build their wobbly sand castles on afterlife forecasts and claims.

While an ungrasping mind can only confront the possibility it may survive death, the grasping mind becomes 100% convinced it can, usually because it appeals to a misunderstood ancient book for which 99.99% is myth, or mistranslated rubbish. (Or purposely re-translated rubbish).

Now, while some certainty may be an assist to many people, the real virulent problems begin when the grasping mind seeks no further than its facile solutions and formulas, because it's desperate to find one that works 100% as an afterlife insurance policy. Thus, the Catholics invent "sanctifying grace" to get heaven bound, while the Christian fundamentalists invent "belief on the Lord Jesus Christ as personal Savior". The true fact is neither of these is an absolute path to the desired goal, since we can't know what - if anything - an "afterlife" means. No one, after all, has actually returned from the literal dead, and told us. And the "resurrection" stories in the New Testament are merely plagiarizations of older pagan God -Man tales (based on Horus, Mithras etc.).

Watts' and the Dalai Lama's crucial point is that the extreme self-centeredness leads one to poison his existing certainty by overtones of anger and attachment.

The process is treacherous and insidious. The mental dynamic works something like this:

"Ok I have found this truth, in my Bible. It says I can lay claim to being saved if I follow this prescription. Now, I am assured of getting into heaven.

BUT, what about all the others, in all other religions, can they also get in by their own routes?

NO, it must not be that THEY can succeed in their way. THEY must follow MY way!

If they refuse they MUST MUST be damned to HELL! There MUST be only ONE path to Truth! And ONE TRUTH! That must apply to ALL and can only be the same as MY way!"


The breakpoint toward anger-attachment arrives with the second line, which is really a question. At some point the believer must ask if others can get to the final destination by their own chosen paths, or if they must necessarily follow his.

Now, he might have said, if a truly liberated being (like the Dalai Lama):"Sure - they can all go their own way!"

The problem is that the grasping mind suffocating in its neurotic need for certainty and pathological fear of uncertainty, can't abide this. Hence, anger and attachment enters. The anger is meted out to all THE OTHERS -who, if they didn't exist, would make life easy for the believer because he'd have only his own way to think about.

So, overtaken by anger (and fear) he's led to falsely reason that if he's right in his path, the OTHERS can't also be right. If he is saved, in other words, THEY must be damned. He can't tolerate them getting into the great place along with him. Or, his beliefs are diluted if he accepts those of the OTHER. The OTHER must therefore be eliminated.

The attachment is by virtue to his own beliefs, without any thought of others'. The (negative-ego based) attachment also poisons his intellect and mind to discrimination between judicious choices as well as moral values. Because he believes (is certain) he has sole possession of truth, he then becomes enabled to think he can do whatever he needs to eliminate the others who dilute it or criticize it. Thus, his morality also prevails over all others, and their moral values must be false or based on the wrong book (source of revelation). Hence, he consigns all other sacred books to the flames, whether Qu'uran, Upanishads, Talmud or other. They must not be granted the same respect as his Bible. And even other bibles are dismissed if they don't meet the exact specifications he (egoistically and falsely) assigns to his.

It was precisely this anger-attachment mindset which has led to the worst genocides the world has seen, because the mass killers believed they were in sole possession of truth.

As the Dalai Lama says (p. 167):

"Our mind is cunning and difficult to control, but if we make a constant effort - work tirelessly with logical reasoning and careful analysis we shall be able to control it and change for the better".

It is interesting to note here the reference to "logical reasoning and careful analysis". In other words, one must operate for the control of one's own mind by more than faith. As Daniel Dennett has noted, (Darwin's Dangerous Idea) "faith" is a meme that is self-protective and resists the scrutiny of logic or analysis. The Faith meme in this sense, is one of the most toxic to logic that can be found. (For example, its standard comeback to a skeptic using reason is that 'faith is superior to reason". But if that is so, faith can never be subjected to rational scrutiny!)

If one desists from using logical reasoning and analysis, then clearly the meme will take over the mind with disastrous results, leading to a self-reinforcing feedback loop. ("I believe because I am RIGHT, I am RIGHT because I believe") The person, rather than being authentic and making authentic choices, then mutates to an automaton under the control of others. This is why nearly one thousand died at Jonestown, Guyana in November, 1978. And that lesson should never be forgotten.

One would hope that one day more religious folks could emulate the Dalai Lama. The problem is that day may come far too late to redound to humankind's benefit.

Looking at Basic Differential Equations (1)

As I pointed out in my blogs on quantum mechanics(QM), differential equations are critical to understanding and using QM. Indeed, the basic foundational QM equation is the Schrodinger equation, a partial differential equation of order two. But, before one can "run" with the higher order partials, one must be able to recognize and solve the humdrum, ordinary linear differential equations. So in the next few instalments (spaced apart!) we will look at different aspects.

As readers may be aware, math is definitely a part of this blog and from time to time I will veer into one or other aspect of mathematics. For those who'd like a compilation of all the math blogs thus far, you can obtain my book: Mathematical Excursions in Brane Space, which is now available at lulu.com. The book is available both in e-format (file download) for $5.00 or as a paperback, 156 page book, for $9.50.

For the sake of reducing the background info, I'm assuming all readers who venture into this area have had at least some exposure to differential calculus. If not, or if you're rusty, you can brush up here:

http://en.wikipedia.org/wiki/Differential_calculus

Let's start our venture into DEs with just about the simplest one imaginable:

dy = x dx

As with all DEs, the solution is accomplished via the process called integration.

(see, e.g. http://en.wikipedia.org/wiki/Integral)

If we integrate both sides, we obtain:

y = x^2/ 2 + c

where c is some undefined (as yet) constant of integration. We call the above the "general solution" to the differential equation. This general solution is, in fact, a family of parabolas. A table of values can also be calculated here to expose the limits for x, y entries. (Readers can do this starting with x = 0 and for corresponding dy/dx (or y'). Thus, for x = 0, y' = 0, and for x = +1/2 (or -1/2) then y' = +1/2 (or -1/2) and for x = +1 (or -1) then y' = +1 (or -1) and so forth. )

What the reader will find when all is over and done with is a graph family such as shown. If we wanted to obtain the particular solution, we'd have to have boundary conditions available. Usually these designate what values x, y are to have at a particular point, and also often the first derivative (y' or dy/dx) at the same point.

Thus, we enter the world of first order differential equations of the first degree.

Let's consider the equation:

xdx + ydy = 0

One could be understandably tempted to write this in terms of the 1st derivative to obtain:

dy/dx = -x/y

but this serves no useful purpose. It's more productive to simply integrate the equation:

xdx = -ydy

directly (since variables are already separated) to obtain:

x^/2 = - y^2 / 2 so

x^2 + y^2 = c

where c is the constant of integration.

Now, the condition is that y = 2 when x = 1 so:

(1)^2 + (2)^2 = 1 + 4 = 5

so the particular solution is: x^2 + y^2 = 5

(Aside question: Can readers figure out why the fractional form x^2 /2+ y^2/2 was discarded?)

Now let's examine a more complex 1st order equation more worthy of the brainpower of the typical readers of this blog:

Find the general solution and the particular curve passing through the point (0,0) of the differential equation:

exp(x) cos(y) + (1 + exp(x)) sin(y) dy = 0

This looks a bit fearsome, but again, the first rule is simplify, which means separating variables (this is also usually where one's acumen with basic algebra comes in!)

we obtain:

exp(x)/ (1 + exp(x)) + [sin(y)/ cos(y)] dy

= exp(x)/ (1 + exp(x)) + tan(y)dy = 0

We then integrate this to obtain*:

ln(1 + e^x) - ln cos(y) = ln c

Or:

ln(1 + e^x) = ln c + ln cos(y)

where again, c is the constant of integration. We can easily simplify the above (using well known principles of natural logs) to get:

1 + e^x = c cos (y)

And this is the general solution.

To get the particular solution we need to substitute the ordered pair values for (0.0) into the general solution, whence:

1 + e^(0) = c(cos (0))

so that: 1 + 1 = c

and c = 2

then we get: 1 + e^x = 2 cos (y)


Now for some problems:

1) Show that y = cx^2 - x is a solution of the DE: xy' = 2y + x

2) Show that y = (2x + c) exp(-x) is a solution of the DE: dy/dx + y = 2 exp(-x)

3) For the differential equation: dy/dx = -x/4y

sketch the curve which passes through the point (1,1)

4) Find the general solution of: sin (t) dp + p cos(t) dt = 0

5) Find the particular solution for the equation:

exp(x) sec(y)dx + (1 + e^x) sec (y) tan(y) dy = 0

and y = 60 deg when x = 3

* (Tables of integrals are often also used to integrate and solve DEs!)

Solutions to the odd ones next time!

Richard Lindzen's Fantasies (2)

We continue to examine Richard Lindzen's recent WSJ tract, 'Climate Science in Denial', on behalf of the flat Earth brigade.

Lindzen’s next bit of balderdash is a 'hit and run' on a letter published in The Financial Times on April 9, written by Prof. Martin Rees and Ralph Cicerone, in which they acknowledge a reduced concern among the public due to "Climategate", in tandem with unusually cold temperatures in North American (there were no such colder temperatures in many other parts of the world – and 2009 again went down as one of the warmest years).

Lindzen skewers the authors for referencing "the uncertainties in feedback effects from water vapor and clouds", adding: “Without these positive feedbacks assumed by climate modelers there would be no significant problem and the various catastrophes that depend on numerous factors would no longer be related to anthropogenic warming”.

But this is bare bollocks. For example, we know that water vapor is not a factor in the initial positive feedback phase and it certainly doesn’t trump CO2. This was pointed out by Spencer Weart in his excellent article: 'The Discovery of the Risk of Global Warming’, in Physics Today, Jan. 1997, p. 34). Weart points out that even a tiny, minuscule amount of CO2 is vastly more efficient at blocking the re-radiation of energy than any amount of water vapor- at those bands. Part of the misconception arose because early researchers, lacking the current technology of infrared spectroscopy, assumed that water vapor bands already blocked out most of what would (ordinarily) be taken by CO2.

Meanwhile, we've known for over 15 years- thanks to climate researcher D. Lubin (Science, Vol. 265, July, 1994, p. 224,) the effects of water vapor only kick in once the super-greenhouse gets under way. Thus, the water vapor will provide a non-linear driver and forcing mechanism to ramp up global warming as the oceans evaporate worldwide. (Evaporation worldwide will commence when the mean ocean temperature reaches 100C, likely in 400+ years now – at the rate of current CO2 absorption and possible future outgassing).

More worrisome, an article in Eos Transactions of the AGU (Vol. 83, No. 34), from August 20, 2002, ‘Progress Made in Study of Ocean’s Calcium Carbonate Budget’, notes that sedimentary carbonates represent the largest reservoir of carbon on Earth. The author also notes that “a third of the anthropogenic CO2 that has been added to the atmosphere since the middle of the 18th century has been absorbed by the oceans”. This means that the oceans, acting as CO2 reservoirs, have actually masked the worst effects of global warming. And that when their saturation point is reached, the spillover effect will be rapid and calamitous indeed.

Other more recent data, including that disclosing increased acidity of the oceans, confirms this. The oceans are now 30% more acidic (because of CO2 absorption) than they were at the dawn of the Industrial Revolution). What does this all mean? It means that the positive feedbacks from CO2 have been making themselves felt on the planet for at least 180 years. Those positive feedbacks have led to a third more anthropogenic CO2 being added, much of which has been absorbed by the oceans – increasing their acidity while concealing the worst effects of direct warming (via the ocean acting as a backup CO2 repository to the atmosphere). So no, the positive feedback here is not “assumed” at all.

What about clouds, cloud cover?

In fact, the relevant positive feedback vis-à-vis clouds was sorted out in the paper: ‘Can Earth’s Albedo and Surface Temperature Increase Together’ in Eos Transactions of the AGU, Vol. 87, No. 4, Jan. 24, 2006, p. 37) As the authors note, though there is some evidence that Earth’s albedo has increased from 2000 to 2004 this has NOT led to a reversal in global warming. (Albedo is a measure of the reflectance of the radiation from the Earth's surface. When one has a higher albedo, then that means less IR should be absorbed and global mean temperatures ought to be theoretically cooler).

The authors cite the most up to date cloud data released in August, 2005 from the International Satellite Cloud Climatology Project (ISCCP). The data – from a range of meteorological satellites covering the entire Earth, discloses the most likely reason for the anomaly is primarily in the redistribution of the clouds.

As the authors point out (ibid.):

“whereas low clouds have decreased during the most recent years, high clouds have increased to a larger extent leading to both an increase in cloud amount AND an increased trapping of infrared radiation”

Obviously then, high clouds count as a positive feedback contributor if they increase trapping of IR. The ISCCP data indicate that high clouds are definitely a players here. Reinforcing this, Prof. Gale Christianson in his book ‘Greenhouse’ (Penguin, 1999, p. 203)has noted that “stratus clouds are gray, dense and low flying and have a net COOLING effect since their albedo is relatively high”.

He goes on to add:

"Conversely, wispy high flying cirrus are semi-transparent to incoming sunlight but block infrared radiation emitted by the Earth thus CONTRIBUTING to the Greenhouse Effect”
Meanwhile, towering cumulo-numbus clouds are neither here nor there and contribute minimally one way or the other." (Ibid)

In other words, Lindzen's "uncertainties of positive feedback" argument is a red herring. We've known which clouds contribute to positive feedback in the CO2 warming scenario, and which contribute to negative feedback, for over a decade. Moreover, the positive feedback effects are gaining as more high clouds form in the atmosphere from a combination of global warming and accelerated evaporation of large bodies of water from heating.

What about other positive feedbacks which can be quantified?

In the first instance, one can regard the critical overshoot of critical CO2 thresholds (e.g. we now anticipate 550-600 ppm is the threshold for the runaway greenhouse effect) in terms of positive feedback, with some gain 'g' - in this case driven by population increase. We know, after all, increasing populations decimate CO2 reservoirs (e.g. forests) even as they pump millions of tons more carbon into the atmosphere each month ....via automobiles, as well as wood burning as fuels.

A rough diagram for some such mechanism can be given as:

(Direct stress)---
!

!
!-------[Direct Mechanism]------>
!
!
!
Gain = g..................................{closed}
!
!
!---------[Feedback Mechanism]---!

By 'total effect' - we mean the CO2 accumulation arising from population growth and its inevitable increased output, from consumption, wastes, fuel use (burning etc.) Where Total effect = Direct Effect x (1 + g + g^2 + ........) = (Direct effect)/ ((1- g) .

If gain g is proportional to population size then: g = k (const.) x population.

For example at a population of ~ 6 billion (6 x 10^9) let the gain factor for increase in CO2 concentration per century be 0.7 (By way of illustration, if the CO2 concentration reaches 550 ppm by 2100, from 380 ppm today, the increase or gain g = (550 - 380)/ 380 = 270/ 380 = 0.71 .

Now, g (6 x 10^9) = 0.7

But: 1/ (1- g) = 1/ (1- 0.7) = 1/0.3 = 3.33

Thus, total effect = 3.33 x (direct effect)

Now, let the population grow to 10 billion, or a 40% increase - which linear relation implies 0.7 will increase by 40% or to 0.98.

Then: g(10 x 10^9) = 0.98

and 1/(1 - g) = 1 /[1- 0.98] = 1/ (0.02)= 50


so the total effect at carbon deposition in the atmosphere is now 50 times greater. In other words, a 40% population increase precipitates a 50-factor or 50/ 3.33 or 1500% increase in response. Of course, if the gain g is nonlinearly related to population increase, the situation would be drastically worse. In any case, we'd be well ensconced in the runaway greenhouse effect, with perhaps 3-4 centuries before the oceans begin vaporizing.

Readers are invited to access Alvin Bartlett's article, Thoughts on Long-Term Energy Supplies: Scientists and the Silent Lie’ by Albert Bartlett (Physics Today, July, p. 53) for further support for the inclusion of population as a direct positive feedback agent in the global warming arena.

More mathematically-inclined readers need to get hold of Bartlett's (2004) monograph: The Essential Exponential! For the Future of Our Planet, paying special attention to Chapter Three: On Population. In it, Bartlett makes an ironclad mathematical case for immediate zero population growth in the U.S. The only reason people resist is: 1) they believe they have a right to make as many babies as they want with no thought to the carbon footprints left by each, and 2) they haven't the foggiest notion how exponential processes, especially consumption of finite resources, work - and the havoc that'll be wrought in the near future. (The most proximate havoc, apart from oil running out and the imminent collapse of our oil-addicted civilization, is the disappearance of adequate water sources).

As for Richard Lindzen, he will still have to pay off a bet he made with one climatologist on the increase in global temperatures. (I'd estimate within about five years!)

Richard Lindzen's Fantasies (1)

Some weeks back (April 22) MIT Prof Richard Lindzen wrote an article appearing in The Wall Street Journal entitle ‘Climate Science in Denial’. He basically argued that, in a nutshell, “global warming alarmists have been discredited”, and the whole global warming science establishment of more than 100,000 was taken down by “Climate gate” the puffed up pseudo-scandal involving hacked emails from The University of East Anglia. My earlier blog on the issue can be found at this link:

http://brane-space.blogspot.com/2009/11/great-e-mail-climate-flap.html


Never mind the hackers committed an act of cyber-terrorism and are still at large, all the focus went to the researchers, such as Michael Mann of the University of Pennsylvania and Phil Jones of East Anglia (director of the CRU or Climatic Research Unit).

But Lindzen is known for writing intellectually dishonest articles in the mainstream media, and this one merely adds to his dossier.

Let’s take first his allegations of “unambiguous evidence of the ethical suppression of information and opposing viewpoints in the emails” and “collusion with other prominent researchers” (by the CRU at East Anglia)

The assertion of ethical suppression is totally specious. Morever, the largest organization of climate scientists in the world, The American Geophysical Union, issued a statement that they found "it offensive that these emails were obtained by illegal cyber attacks and they are being exploited to distort the scientific debate about the urgent issue of climate change."

They also reaffirmed their 2007 position statement on climate change "based on the large body of scientific evidence that Earth's climate is warming and that human activity is a contributing factor. Nothing in the University of East Anglia hacked e-mails represents a significant challenge to that body of scientific evidence"

Interested readers can find that statement here:

http://www.agu.org/sci_pol/positions/climate_change2008.shtml

Lindzen's assertion of a collusion is nonsense. The vast majority of the emails related to four climatologists: Phil Jones, the head of the CRU; Michael E. Mann of Pennsylvania State University (PSU), one of the originators of the graph of temperature trends dubbed the "hockey stick graph";Tim Osborn, a climate modeller; and Mike Hulme, director of the Tyndall Centre for Climate Change Research. The four were either recipients or senders of all but 66 of the 1,073 emails, with most of the remainder of the emails being sent from mailing lists. A few other emails were sent by, or to, other staff at the CRU. Jones, Briffa, Osborn and Hulme had written high-profile scientific papers on climate change that had been cited in reports by the IPCC.

Evidently not on Lindzen's radar (conveniently) East Anglia announced on March 22nd the composition of an independent Science Assessment Panel to reassess key CRU papers which have already been peer reviewed and published in journals. The panel didn't seek to evaluate the science itself, but rather whether "the conclusions reached by the CRU represented an honest and scientifically justified interpretation of the data." The university consulted with the prestigious Royal Society in establishing the panel. It started its work in March, 2010 and released its report on April 14. During its inquiry, the panel examined eleven representative CRU publications selected by the Royal Society that spanned a period of over 20 years, as well as other CRU research materials. It also spent fifteen person days at the UEA carrying out interviews with scientists.

The panel’s conclusions confirmed that nothing in the emails proved wrongdoing, and dismissed the allegations. Independent reviews by FactCheck and the Associated Press said that the emails did not affect evidence that man made global warming is a real threat, and said that emails were being misrepresented to support unfounded claims of scientific misconduct. They also concluded that there were disturbing suggestions that scientists had avoided sharing scientific data with sceptical critics.

Penn State reviewers, in their (independent) inquiry into allegations of misconduct found: “the so-called 'trick' was nothing more than a statistical method used to bring two or more different kinds of data sets together in a legitimate fashion by a technique that has been reviewed by a broad array of peers in the field.” The Parliament of the United Kingdom select committee inquiry concluded that : "the 'trick' appears to be a colloquialism for a "neat" method of handling data," and "[hide the decline] was a shorthand for the practice of discarding data known to be erroneous."

As noted in a New Yorker sidebar piece on the findings, the climate emailers, Mann et al, were mainly taken to task for being careless with their use of language as well as with their computer security. They were not upbraided or disciplined for any "shenanigans" as Lindzen's febrile imagination appears to imply. All in all, as the New Yorker noted, it was all a "tempest in a teapot" designed by the denier -Flat Earth battalion to build up pseudo-morale for its troops, and again - try to sway the public mind.

Indeed, the statistical device employed by Mann was no where near as egregious as the tactic employed by Willie Soon and Sallie Baliunas (two well known deniers) in a 2003 paper published in Climate Research- a known denier journal. (One of the journal's editors, Chris de Freitas of the University of Auckland, has frequently editorialized in the New Zealand press against the overwhelmingly accepted conclusions of the IPCC.)

In the case of Soon and Baliunas, they simply excised quality data at higher resolution (30 –year intervals) to conceal the emergence of a warming signal, so their paper could ignore it. Their choice of using 50-year data increments (when the IPCC scientists already disclosed anthropogenic warming appears at 30 -year levels), would have seen the paper quashed in any proper peer review, by any referee remotely knowledgeable of statistics. In effect, Baliunas and Soon exploited what we call a 'selective effects filter' to exclude ab initio the data they preferred not to address. But did any brouhaha erupt on the scale of the bogus climate email flap? Not on your life! It never even reached the radar of the mainstream media.

Lindzen’s next bit of boneheaded nonsense emerges when he writes:

"the (IPCC) observations are consistent with models only if emissons include arbitrary amounts of reflecting aerosols, particles (arising for example from industrial sulfates) which are used to cancel much of the warming predicted in models"

But he has this exactly backwards. We've already known for decades about the degree of the ambient cooling factor from these pollutants, which were originally investigated by Gerry Stanhill, an English scientist working in Israel. Prof. Stanfill compared sunlight records from the 1950s with current ones, and was astonished to find a 22% drop in solar radiation. Intrigued, he searched out records from all around the world, and found thesame story almost everywhere he looked, with sunlight falling by 10% overthe USA, nearly 30% in parts of the former Soviet Union, and even by 16% inparts of the British Isles.

Stanfill called the effect "global dimming" in research published in 2001, and of course met with a barrage of skepticism. The reason is that validation of his work would confirm that aerosols and sulfates are already accounted for in the "arbitrary coolants" factor balance sheet for climate change - but not in a way desired. That is, since major pollutant reduction via legislation such as the Clean Air Act in the early 70s, it means that global warming has actually been concealed by a factor of up to 30%. In other words, Lindzen's arguments are turned on their head: the reduction of aerosols and sulfates in the atmosphere since the 70s means that now warming is ramping up with the dimming factors abated.

As things stand, CO2 levels are projected to rise strongly over coming decades, whereas there are encouraging signs that particle pollution is at last being brought under control. According to Dr Peter Cox, one of the world'sleading climate modellers.:

"We're going to be in a situation, unless we act, where the cooling pollutant is dropping off while the warming pollutant is going up. That means we'll get reducedcooling and increased heating at the same time and that's a problem for us".


Contrary to Lindzen's foolishness, this implies that even the most pessimistic forecasts of global warming may now have to be drastically revised upwards. That means a temperature rise of 10°C by 2100 could be on the cards, giving the UK a climate like that of North Africa, and rendering many parts of the world uninhabitable. That is unless we acturgently to curb our emissions of greenhouse gases.

Next: Lindzen's "Positive Feedback" Foolishness.

The Paradox of Federal Agencies

In the last months many Americans may well wonder what's been going on at numerous Federal Agencies charged with oversight of key sectors. Whether that be the Minerals Management Service (M.M.S.) which inappropriately approved deep sea drilling using a non-standard 7" diameter pipe as opposed to the 10" norm, to the SEC, which somehow blew it during the Bernie Madoff Ponzi scheme.

As the news unfolded, we learned that key personnel at each Agency were whiling away the days gazing at Internet porn. It became something of a joke, albeit not to those stakeholders who've been punished severely on account of the lax regulation.

But the paradox of federal agencies is why some are so ineffectual, while others have risen to the challenge (such as the EPA, IRS) and certainly in relative terms.

In a recent New Yorker piece ('The Regulatory Crisis', June 14 & 21), it's acknowledged that both the SEC and M.M.S. were the pits. The author notes, for example, how the M.M.S. "allowed Massey Energy to flout safety rules", while the financial regulators "let A.I.G. write more than a half a trillion dollars of credit-default protection without making a noise". (See my previous blogs on credit default swaps from last year, especially).

Meanwhle, the SEC "failed to spot the frauds at Enron and WorldCom and gave Bernie Madoff a clean bill of health." (Well, we're aware in retrospect that one of the SEC regulators had been engaged to Madoff's niece. Maybe it played a part, maybe not. Or maybe it was too much obsession with porn by lower echelon personnel!)

Of course, the usual culprits are cited, especially the notorious "revolving door" between government and industry - which enables the classic set piece of (industry) "foxes" guarding or overseeing government "henhouses". This isn't limited to regulation by the way, it extends to the pernicious and cancerous lobbyist parasitism that uses lobby-owned or directed mavens (like Bill Tauzin, R- La. in 2003 with The Medicare "Modernization Act") to craft legislation friendly to industry. (That "doughnut hole" didn't occur by coincidence, nor did the inability of Medicare to bargain for the best drug prices, like the Veteran's Administration!). Once Tauzin departed the House, he found his way right back to PhrMA as a prime lobbyist)

As I've noted before, in other nations - like Barbados- once a person serves in government he must wait at least ten years to return to serve or run in any office, if he's been in an industry that government depends on (e.g. Oil, Utilities, etc). That is, he or she must wait 10 years after leaving the industrial post before entering government service. Harsh yes, but then Barbados has now preserved a genuine democracy for over 370 years, as opposed to enabling a corporatocracy to hatch and for corporations to trump the votes of citizens.

Anyway, the author addresses two sources of the problem, with which I agree:

- Regulation (at least since the Reagan era) has been de-legitimized and is "seen as little more than a tool of Washington busybodies"

- Congress tends to take a "famine or feast" attitude to agency funding, allocating more when times are good, and much less when not. It also funds according to the party in power, controlling congress. For the consistently "de-reg" repukes, that means a subsistence budget and nothing more.

The last is wildly divergent from what governments in Europe do. They regard the protection of citizens' interests as far too important to leave to the vagaries of political cycles and concomitant income inflows. They instead allocate an appropriate initial set rate which can go up as more regulatory crises appear (such as the recent one to do with credit default swaps).

Other astute nations (e.g. Singapore) that respect the commonweal and regard it as the primary support of a society (as opposed to the military-industrial complex) reward regulatory agencies with salaries competitive with those of industry. That these nations are amongst the least corrupt in the world, with the most efficient bureacracies, is therefore not surprising.

Meanwhile, we have under-funded agencies in this country, like the SEC and M.M.S. (which was originally the domain and creation of Reagan's Arch Christianoid Interior Secretary James Watt- who once opined that the forests, resources of the country are exploitable given that Man "has dominion over them"). With little funding there's no scope for actual regulatory oversight, and too little manpower to make a difference, not to mention political players looking askance at anyone interfering in their "games". No wonder the regulators in these low-funded and marginalized agencies are reduced to porn-watching. But hey, at least the dopamine secretions in response prevent them from going "postal".

Meanwhile, citizens can applaud that the IRS and EPA are garnering respect and their personnel can walk with heads held high. The IRS continues to press its case against the Swiss Bank UBS to obtain the names of thousands of U.S. taxpayers who've been hiding their millions there. No political toady in this country has demanded the IRS back down, which shows me their personnel have a measure of self-respect in what they do. They know they are pursuing a proper goal so they can't be compromised. The same now also goes for the EPA which has pressed forward on a number of fronts, including recently exposing the toxic stew of 80,000+ chemicals that flood our ecosytems- many of them cancer causing agents (first pointed out by Devra Davis in her must-read: "The Secret History of the War on Cancer") while also making aggressive provisions to combat global warming if congress fails to act.

One thing I'll wager for sure, you'll not find IRS or EPA personnel wasting time watching porn. Their agencies are properly funded (relatively) and that indicates they're empowered to operate in good faith rather than bad faith. To Existentialist Jean -Paul Sarte, bad faith was like a cancer devouring a person's being, continually lowering its value and esteem. A person was disabled by whatever external circumstances to meet the standards of his job, or forced in some other way to live inauthentically. (Note , however - even the most suppressed mouse in an agency can still be a whistle -blower!)

What we need is to shore up the weakened federal agencies across the board, and oust all the political appointees thwarting their main missions. Then, we need to allocate funding, including competitive salaries, like other advanced nations do.

If we fail to act, we can't be surprised when the next salmonella outbreak occurs with hundreds felled, or the latest financial scam collapses with billions lost via a "product" unheard of hitherto.