Solutions To Clairault DE Problems Revisited

  1) Solve:  p² x - y  = 0

Solution:  

Solve for y:  

y =  p² x

Differentiate:

dy/dx = p =  p² + 2px (dp/dx)

Re-arrange:  

2 dp/(p -1) + dx/ x = 0  

Solve: 

(p - 1) ²  x = c  

Eliminate p between previous two eqns.  

y - 2Öx  + x =  c  

Which is the general soln.

2) Solve: p² + 2y  - 2x  = 0

Solution: 

Solve for x: 

x=  ½(2y + p²) 

Differentiate:

dx/dy = 1/p = 1  +  p (dp/dy)  

Simplify:

p² dp/  (p - 1) + dy = 0

=>  [p + 1 +  1/ (p-1)] dp + dy = 0  

Solving:  

½ p² + p  + ln (p - 1) + y = c

Which is the general solution.  The parametric form of the general solution is: 

y = c - ½ p²  - p  - ln (p - 1)

x = c  - p  - ln (p - 1)

Media Disparaging Of Joe Biden Continues With WSJ Troll Not Fit To Shine His Shoes

                           Trump's psychopathic mugshot: $7 million raised already 

"I happen to think that Democrats would be safer with a nominee who’s younger than Biden is and radiates more energy than he does. But I believe at least as strongly that if the unideal choice before Americans winds up being Biden, with his imperfections, or Trump, with his, rejecting Biden because of how old he has grown isn’t a grown-up decision."- Frank Bruni, NY Times, 'Trump Is Really Old Too'

I suppose I shouldn't have been surprised when not long after Trump and 18 members of his cabal were booked last week, with mugshots taken at the Fulton County jail, yet another Biden media takedown effort would soon follow. In this case it came from a little Aussie turd named Gerard Baker who is described as "editor at large" at the Wall Street Journal. Well from his attempted demolition piece on Monday, The Media Try to Inflate Biden’s Stock,  under which one read:

Journalism by omission aims to suppress news of the president’s frailties and possible misdeeds

There was little doubt it was more a troll piece than truth piece. It was aimed at further tarnishing Biden's successful brand and accomplishments before the upcoming '24 election.  The motive? If Trump is gonna go down under the weight of 91 felonies and 4 indictments, we're gonna take Biden down too.    So it's not a case of the media "inflating Biden's stock" which they certainly have not done the past 2 years, e.g.

Why Are Biden's Poll Numbers So Low Despite Vast Personal Satisfaction... 

But rather the rodents occupying the rat warrens of the WSJ trying to tear down Biden's image further, reducing him to a doddering, drooling doofus who can't find his shoes for his sleeves and could never have crafted the successes the Dems claim including:

i) Inflation went down after passage of the inflation reduction act and is now near 3.2%. We actually have the lowest inflation amongst the G7 nations (See Harmonized Headline HICP graph).

ii) U.S. has had the highest GDP growth in the G7 group (Again check the available graphs!)

iii) Unemployment in May was lowest since 1969, and the lowest income workers made the largest historic gains.

iv) Housing starts have surged to the most in 3 decades (Reuters)

Less noted: 

v)The typical U.S. worker produces nearly 5x the output of a Chinese worker. Also, American worker output is greater than that of Europeans, Japanese, British, Canadians and Australians by a wide margin.

vi) The U.S. arguably has the deepest and most liquid financial market in the world enabling the birth of new businesses and continued growth of successful ones. The public financial value – known as market capitalization- now stands at 170%. (For most countries it is below 100%)

According to the UK magazine The Economist: “By a whole range of measures, American dominance is striking”.

So by any logical measures Biden indeed has one of the best narratives in terms of economic success. Despite all the disruptions and headwinds the U.S. has the best economy in the world. Yet the Reeptards are determined to bring him down, by piffle such as Baker put out on Monday.  Why? Because for his traitor psycho hero Trump to have any chance at all, he and his fellow WSJ op-ed trolls must seek to make Biden appear weak and incompetent. And indeed he has succeeded amongst the clueless WSJ rightist readers who wrote the following comments;

Thanks to the WSJ Op/Ed section to at least bringing a semblance of coverage of the liberal media campaign!

Biden has wrecked the country already! Keep digging up more dirt!

You can make Biden blink, but you can't change the price of gas or groceries. Eventually the economy is good and the world is a safer place or not.

Unfortunately people will have had to suffer four years of bad domestic and foreign policy before they realize that he was asleep on the job!

Baker began by focusing on Biden's Maui trip and the 80 yr. old president sympathizing with locals' post -disaster plight. Something Trump would never do (he'd more likely toss paper towels at them like he did in Puerto Rico after it was struck by a hurricane during his term).  Baker also brought up the trope of "Sleepy Joe" and that Biden may have been blinking, because he was partly asleep during his meet and greet with Lahaina locals.  This led Baker to pounce, writing:

"A less partial observer might conclude that if we need high-definition video to be certain that the president’s eyes are “open, blinking” while he performs his duties, we have a problem more serious than mean-spirited conservatives making fun of him."

But he wasn't done, then trying to blame the mainstream media for providing cover for Biden's flaws, foibles, verbal gaffes etc.:

"Media bias is nothing new, but this goes further. For almost four years there has been a concerted, sustained and so far successful effort by supposedly independent media organizations to elect, defend and preserve in office the nation’s leading Democrat. It is immanent throughout the coverage of this president and mostly takes the form of misrepresenting reality by willfully ignoring or suppressing anything that undermines him—what we might call journalism of omission.

The juxtaposition of the attention given to the Hawaiian fact-check with the effort to look past the many infelicities of the president’s island misadventure is a small example of the Biden-Keeping Operation."

"Island misadventure"? WTF are you bellyaching about? But you can't make this blowhard journalistic bombastic bullshit up. It is now the warp and woof of the reactionary Right media, especially at FOX and the WSJ op-eds. But Baker is cockeyed foolish if he thinks the mainstream media have been giving Biden a pass for 3 years,  given they've relentlessly pounded his age, e.g.

• Biden's Age - A Problem for the Media - But Not Sensible Citizens...

So why so much negativity? The Economist’s assessment of the data suggests the pessimism we’re seeing is nothing more than “emotional baggage brought on by a perpetual negative news cycle”.    This emphasis is because newsrooms are convinced that only negative material, content gets clicks or reads.  So beating a president who's the polar opposite of the feral wretch Trump is cool, especially over his age, inflation and other factors beyond his control.   Like the price of gas at the pump. 

This ongoing litany of whining leads to an entrenched negative news cycle and pessimistic information bias. Much of it is driven by politics (often from the Trumpian Right) working overtime to decapitate Biden and the Dems and thereby turn lazy people into automatons who will vote against their own interests. Why? Because the tropes and canards are then regurgitated on social media and the dregs of the net like 4chan, 8chan, Telegram etc.  

As Chris Hayes pointed out in an ALL In episode 2 weeks ago, the Reeps are going deep in to pile on Biden because it provides them with an excuse not to admit their traitor hero is a piece of degenerate slime, a pestilence parasite and a malignancy on the Republic. A walking turd who ought to have been put away multiple times over by now. As opposed to still being out and about, spewing all manner of vile rhetoric about the judges, the prosecution and especially Jack Smith.  E.g.

Deranged Imp Trump Needs To Be Locked Up For His (and OUR) Own Good 

Any other person would have been remanded to custody long ago, but this roach is allowed to keep on keeping on with his toxic torrents which - as former prosecutor Cynthia Alksne noted two nights ago- could well have severe repercussions for a DA, judge or prosecutor. But despite that, no one is willing to take the orange fecal fungus down.

See Also:

|Opinion | Can states remove Trump from the ballot? Yes, they might. - The Washington Post

And:

Donald Trump

Trump’s True Believers Bring the War on Terror Home

by Norman Solomon | August 30, 2023 - 5:57am | permalink

Ever since Donald Trump became a former president, news outlets and commentators have cited polls showing that many Republicans believe violence might be needed to save the country. As Trump’s legal woes increase, so do mainstream media alarms about the specter of violent responses. But we’ve heard virtually nothing about connections between two decades of nonstop U.S. warfare overseas and attitudes favoring political violence at home.

And:

Donald Trump

Confessions of a Zealous Trumper

by Robert Becker | August 28, 2023 - 5:29am | permalink

Will stupid leftists ever behold
Why MAGA captures young and old?
Deadbeat libs troll our sainted hero,
But who’d make a better pharaoh?

Unshakable fans know far better –
Bigly causes surpass the plate-setter;
The left guffaws at the “Trump cult” –
Yet sad envy drives that cheap insult.

Crackpot libs can’t match our grievances
Because they lack real allegiances;
Only patriots with righteous causes
Defy bad laws, like daring outlaws.

Revisiting Differential Operators.

 Differential Operators are basically shorthand ways of writing derivatives and by extension writing more concise differential equations.  In general:

(D _x y) ⁿ  =   dⁿ y / dxⁿ       n=  1, 2, 3….

Then:  (D _x y) ²  =   d² y/ dx²      

 

(D _x y) ³  =   d³ y/ dx³      

And so forth.

For example:  d⁴y/ dt⁴

Can be rewritten: (D _t y) ⁴ 

And:   2d³y/ dt³  

Can be rewritten:  2(D _t y)³ 

Then rewrite the differential equation:

d⁴y/ dt⁴ – 2d³y/ dt³  –7 d²y/ dt²  + 20  dy/dt  –12 y = 0

Using differential operators. 

Using the preceding hints for the notation, we have:

(D _t y) ⁴ -  2(D _t y)³  - 7 (D _t y) ²  + 20 (D _t y) - 12y = 0

Of special interest is the inverse differential operator, viz.

(D^-1  x ⁿ )   =    ( xⁿ⁺¹ )  /  (n + 1)

Similarly for higher order::

(D^-2  x ⁿ )   =  (D^-1 ) ( xⁿ⁺¹ )  /  (n + 1) = 

xⁿ⁺²  /  (n  +1 )  (n  +  2) 

And for higher order (k) in general:

(D^-k  x ⁿ )   =    (x^n+k )  /  (n+ 1) (n + 2)...(n  +   k)  

Thus,  1/D stands for the integral, e.g.   ∫  F(x)dx  but with denominator corrected for as given by inverse formulae above.

And 1/ D ⁿ  stands for successive (n)  integration.

Suggested Problems:

1)  Rewrite the DE below with differential operators:

   5 d⁵y/ dt⁵ - 10 dy/dt –25y = 0

2) Rewrite  the DE below in standard derivative form and solve:

(D _x y)   =   3x²   -  1  

3)  Does  (D _x y) ²  =   D ² _x y  ?  Explain.

4) Evaluate each of the following:

i) (D _x y)  (ln e ^x / 1+  e ^x)

ii) (D _x y)  ½ ( e ^x   -  e ^-x)   

iii) D^-1  ( 11 x² )  

 iv)  (D _x y)    ( e ^x   ln x)

InSight Mars Lander Uses Doppler Effect To Discover More Rapid Mars' Rotation

 

                              Mars, now a mystery planet with a mystery rotation

                               Key component of Mars InSight Lander (JPL/NASA)

The question in the wake of a paper published in June in the journal Nature is: Why is the rotation of Mars speeding up?  NASA’s InSight lander fell silent in December after dust accumulated on its solar panels but new data have revealed surprises.

To track the planet’s spin rate, the study’s authors relied on one of InSight’s instruments: a radio transponder and antennas collectively called the Rotation and Interior Structure Experiment, or RISE. The technique for measurement entailed beaming a radio signal to the lander using the Deep Space Network. RISE  then reflected the signal back. When scientists received the reflected signal, they then looked for tiny changes (variations) in frequency caused by the Doppler shift (the same effect that causes an ambulance siren to change pitch as it gets closer and farther away). Measuring the shift enabled researchers to determine how fast the planet rotates.

The researchers found the planet’s rotation is accelerating by about 4 milliarcseconds per year² – corresponding to a shortening of the length of the Martian day by a fraction of a millisecond per year. I.e. a more rapid rotation rate. This is a subtle acceleration, and planetary scientists aren’t entirely sure of the cause. But they have a few ideas, including ice accumulating on the polar caps or post-glacial rebound, where landmasses rise after being buried by ice. The shift in a planet’s mass can cause it to accelerate a bit like an ice skater spinning with their arms stretched out, then pulling their arms in.

 The paper examined data from InSight’s first 900 Martian days – enough time to look for such variations. Scientists had their work cut out for them to eliminate sources of noise that could produce similar variations: Water slows radio signals, so moisture in the Earth’s atmosphere can distort the signal coming back from Mars. So can the solar wind, the electrons and protons flung into deep space from the Sun. So can the relative positions of Earth and Mars in their respective orbits.

Once planetary scientists accounted for these ancillary factors there remained leftover frequency variations, suggesting an enhanced rotation. In particular, the additional findings gleaned from tiny shifts in the frequencies of radio transmissions between Earth and InSight on Mars.

RISE is part of a long tradition of Mars landers using radio waves for science, including the twin Viking landers in the 1970s and the Pathfinder lander in the late ’90s. But none of those missions had the advantage of InSight’s advanced radio technology and upgrades to the antennas within NASA’s Deep Space Network on Earth. Together, these enhancements provided data about five times more accurate than what was available for the Viking landers.

According to the paper’s lead author and RISE’s principal investigator, Sebastien Le Maistre at the Royal Observatory of Belgium:

"What we’re looking for are variations that are just a few tens of centimeters over the course of a Martian year. It takes a very long time and a lot of data to accumulate before we can even see these variations.”

Adding:

"Mars, because it is not a perfectly round sphere, wobbles like a top. but the primary goal was to measure the rotation,” 

 An additional related bonus is that precise measurements of the rotation (with all of the wobbles) places constraints on the structure and composition of the very deep parts of the planet, according to Dr. Le Maistre.

Similar measurements had been attempted during NASA’s Viking missions in the 1970s and also during later missions like Pathfinder in 1997, but those were not precise enough or long enough. To show how unique, Le Maistre emphasized: 

“This was never done for any planet other than the Earth before,”

From the magnitude of the wobbles, the Le Maistre's team has calculated that the molten core of Mars is about 2,280 miles wide. (Mars as a whole is about 4,200 miles in diameter.)

See Also:

Mars is spinning faster, here's what it could mean | This World - YouTube

FT Columnist Claims "Surplus Mental Capacity" Fuels Conspiracies & Psycho Babble? Give Me A Break!

 

                                         FT's  Janan Ganesh - Overthinks basis of delusions

                                Alex Jones: No example of 'cognitive overcapacity'

Financial Times political columnist Janan Ganesh has often sought to be provocative in his various opinions, and in most cases met at least minimal standards of rationality and attention to reality.   But in his latest column ('The Age of the Clever Fool') he appears to have gone off the rails by way of overthinking. 

In the lengthy subheader we see he scribbles:  "Tech bros, woke theorists, psychobabblers - George Orwell endures because his plainness is relief from all".

Adding: "The problem traverses the ideological spectrum. The movement known as woke could only have been incubated and hatched on campuses. Few places have so much intellectual potential. Few places have so little, well, I am going to call it “lived experience”.

His thesis based on these preliminaries is basically that the explosion of bunkum across the ideological spectrum - from Covid and vaccine denial, to climate denial, to climate doomism to mindless political populism (Trumpism), to general conspiracy ideations - hatched from QAnon or RFK -  are all of a piece, namely neural glitches in the modern world.  They boil down to fads, conspiracies, psychobabble, self-help mumbo- jumbo etc. all of which "might be explained as surplus mental capacity casting restlessly around for some outlet. "

Basically, noting that scientist Peter Turchin "has argued that underemployed graduates are potential social trouble". But what about the millions of kooks and conspiracy freaks  - like Alex Jones - who aren't college grads?  How come they exert so much power given they have minimal surplus mental capacity?  Ganesh has no answers because it's much easier to just invent a facile trope and try to embellish it to the point some may be snookered. 

Yes, a fair number of university -educated people did fall for the promises of the likes of Sam Bankman-Fried with his crypto investment offers, and Elizabeth Holmes with her "thumb prick" blood drop tests.  Hell, I even fell - somewhat - for the latter, or at least hoped there was some ballast to the idea, given I was in the midst of multiple rounds of blood tests (psa, testosterone, alkaline phosphatase) concerned with my cancer. But it was not to be, and I quickly realized I'd been had so had to stiffen myself to submit to future rounds of blood tests of the regular kind.  So Ganesh is correct up to a point. But he over-generalizes and exaggerates.  

Admittedly, he identifies mucho bunkum and just a fraction is enough to enrage the confirmed rationalist.  Like the "effective altriusm" thrust which purports to help others up to a point and which purpose is embodied in the principles of its website:

CEA's Guiding Principles | Centre For Effective Altruism

Then there is Elon Musk's own program to help multiply the super-gifted by selective reproduction: namely his own. E.g.

We need to talk about Elon Musk's breeding program (inverse.com)

You can't get much more top heavy with crank-hood than that. But close to it is the nest of longevity research cranks and kooks who believe human life can be endlessly extended and also it is worth it to do such, e.g.

Transhumanism : Another Cockeyed Concept Destined To Fail From Human Hubris

As I noted in that post:

"Now an even more cockeyed immortality denizen has entered the picture, called the "transhumanist"  (WSJ,  June 20,  'Looking Forward to the End Of Humanity')  This lot don't merely expect to conquer old age but to surmount the entire spectrum of biological fragility, to end up with a kind of "transhuman future".   Thus (ibid.):

"With our biological fragility more obvious than ever, many people will be ready to embrace the message of the Transhuman Declaration, an eight-point program first issued in 1988", e.g.

https://humanityplus.org/philosophy/transhumanist-declaration/

And what, pray tell, are we looking at here?  Well, among the "avenues to immortality" discussed are:

-  Creation of nanorobots which could be programmed to live inside our cells and constantly repair any damage, halting aging in its tracks.

-  Genetic engineering could eliminate the mechanism that causes us to age in the first place. (And could also deliver genetic 'mistakes' such as hybridomas, or else humans with serious genetic defects.)

- Transferring consciousness to special computers - where it "can survive indefinitely".

Where Ganesh runs into trouble is when he over-extends the basis of his perspective on "clever fools" and their "surplus mental capacity" to examples where these need not apply.  As when he writes:

"When the cultural left makes contact with that less credentialed but longer-in-the-tooth movement called the electorate, see how it struggles. 

Those dupes in turn can point to the mass market for amateur psychotherapy and self-help mumbo jumbo. And so the carousel of eloquent, educated naïveté goes round. There have always been clever fools: Cambridge spies, eugenics-smitten Fabians, Hitler appeasers with All Souls fellowships. During his leadership of the Pentagon, Robert McNamara, the ultimate genius/sucker, fought the Vietnam war as a hedge fund quant might. The problem existed in pockets back then, however, because so few people went to university. Now it is almost ambient."

But can we lay all of that at the feet of university education?  What about the millions who've graduated - from thousands of universities globally the past 50 years- and aren't caught up in crank conspiracies, New Age psycho-babble, eugenics obsession, vaccine rejection, climate denial or other mental flotsam and jetsam? There has been a substantial mass of graduates - I'd say a preponderance- whose brains haven't been colonized or seeded by such mind viruses because they've been immunized by critical thinking. For that, yes, I have the Jesuits to thank at Loyola University, for opening the Loyola Fieldhouse - in January, 1965, to atheist existentialist Jean -Paul Sartre, e.g.

To debate the (then) recent Catholic convert philosopher Gabriel Marcel.  The debate was totally engrossing and lasted for 90 minutes, keeping the 900 or so students (from Tulane and Loyola) spell bound. It was how I first became interested in existentialism even buying Sartre's masterpiece 'Being and Nothingness'  (available at the Loyola Bookstore):

Sartre's masterpiece which I still have

True, we have lived through a tremendous expansion of cognitive training (via university education) and we're told the current knowledge - just in science doubles ever 3 years.  Look at the explosion of just astronomical journal papers, only a minuscule fraction of which actually get read. To get an idea of the sheer expanse of pure research-  merely  in astrophysics - check out the papers in the link to  this Astrophysical Journal issue below:

• Number 2, 2018 February 1

But in critical thinking application the discipline of thought is a prerequisite so one does not "go off the rails".  But this is what's missing in most university education now. Ganesh in his essay points to the absence of "a commensurate increase in that unteachable trait known as our judgment".   But this is part and parcel of what I call disciplined thought editing in critical thinking application. It is what, indeed, cannot be taught though it can be assimilated through much arduous trial and error.  In time, then, one becomes a rigorous editor of one's own thoughts and recognizes which fit in a coherent milieu and which do not. But if one is overpowered, say by any kind of new knowledge, this may not be feasible.

In that case, yes, the "law of unintended consequences forever applies". Then Peter Turchin could argue that "underemployed graduates are potential social trouble."  Why? Because they are generally ensconced in work or other environments  (FOX News) wherein neither critical thinking or disciplined thought is required.  So anything goes.

Why would that situation be much less true of the most successful graduates? To embrace a dangerous dogma, whatever it is, probably takes some resentment, but it definitely takes some grounding in conceptual thought.  I suppose I embraced Sartre's atheistic existentialism after Loyola, and yeah, maybe resentment at my Catholic upbringing played a role. But the grounding in conceptual critical thought that Loyola's Jesuits provided me, allowed me to coherently develop my atheism leading to my series of atheist books:

Ganesh insists a "vast minority of the public now has it" (surplus mental capacity), which is true.  But I dispute that vast minority has the necessary critical thinking skills and discipline of thought to manage that cognitive surplus capacity.  Else they would actually DO something constructive with it.

See Also:

What's Behind The 'Critical Race Theory' Hysteria?

And:

Enough Of The Anti- Woke Bunkum - The Culture Warriors Remain On The Right 

And:

The Intelligent Irrationalists Revisited

And:

A Misplaced Emphasis May Be Why A Majority Of Respondents In WSJ Poll Say College 'Not Worth The Cost' 

And:

Dispatching Elon Musk's Fairy Tale Of A 'Woke Mind Virus'..

And:

• Let's Admit It: If You Succumb To "Fake News" You're A Dummy ...

And:

• WHO Are The Crazies Disrupting The Country?

And:

Why do Republicans disproportionately believe health misinformation?