Solutions To Laplace Transform Problems

 1) Solve:    

d ³ Y/ dt ³ -    d Y /dt   =   0 

Using the Laplace transform and the conditions:

Y(0) =  1,  Y’(0) =  0  and  Y’ (0) = 1

Solution:

We write:

£ { d ³ Y/ dt ³  } =  s ³   y(s)  -   Y(0) s ²    -    Y'(0) s  -  Y"(0)


=    s ³  y(s)  -   s ²    -   1   


£ {dY / dt}   =    s y(s)   -  Y(0)  - Y"(0)


=    s y(s) -  1   


Substitute the expression for each transform:

s ³  y(s)  -   s ²    - 1   =   0 


Collect like terms and transpose:

  (s ³  -     s)  y(s)   =      s ²    +  1  

 
Solve for y(s):

y(s)   =   ( s ²    +  1  ) /  (s ³  -  s)  =    ( s ² +  1  ) /  s (  s ²    -  1   ) 

Use  of partial fractions yields:

A/ s   +    B/ (  s ²  -  1   )     =   s ²    +  1  

Then:

 A (  s ²  -  1   )   +  Bs   =   s ²    +  1  

Whence:

A s ²     -  A    +  B s    =   s ²   +   1


Yielding values (by equating coefficients):

A    = 1,    B  -  A   =  0,  

So:  B  =    1

è

( s ²  +  1  ) /  s (  s ²    -  1   )  =  1 / s     +   1 /  ( s ²    -  1   )


We then need to take the inverse Laplace transform : 

 £ ^-1  [1 / s     +   1 /  ( s ²    -  1   ) ]

=  

£ ^-1  [1 / s ]    + £ ^-1 [ 1 /  ( s ²    -  1   )]   

From the table of transforms (Feb. 7 post) :

We see:   1/s   is inverse transformed to  1

And similarly:  [ 1 /  ( s ²    -  1   )]  ->   cosh (t)

 
Then the solution to the DE is:  

Y (t)   =   1   +    cosh (t)

2)   Solve:    

  d² y/ dt ²  +  4y    =   3 sin t 

 Using the appropriate Laplace transform:

With y    =  F(t),   F(0) = 1, F'(t) = 0

  

Solution:  

Write:

 £ {F’’(t) }  + 4 £ {F (t) }  =  £ {3 sin (t) } 

 £ {F’’(t) }  + 4 £ {F (t) }  =    3 / s ² +  1

 £ {Y’’(t) }   =    s ²  F(s) -   s 

4 £ {F (t) } =  4 F(s)

Then: 

 F(s) = s ²  F(s) -   s  + 4 F(s)  = 3 / s ² +  1 

Using partial fractions we get:  

F(s) =  3 / (s ² +  1) (s ² +  4)  +  s /(s ² +  4)      

=  1 / (s ² +  1) -   1/ (s ² +  4)  +  s /(s ² +  4)

After taking inverse transforms:

F(t) =  sin t  -  ½   sin 2t   + cos 2t

All Experts Redux: Can One Abolish Kepler's 2nd Law Of Planetary Motion?

 Question:  If we relocated the Sun in one focus of the orbit - say from the diagram in the link below

http://starchild.gsfc.nasa.gov/docs/StarChild/solar_system_level2/activity/song.html

we will discover that the straight line between the Earth and the Sun sweeps the interval of time between March 21 and Jun 21 is much bigger than that swept between September 21 and December 21, and the distance crossed by the Earth from March 21 to Jun 21 is also bigger than that crossed from September 21 to December 21.  This is  contrary to Kepler's second law so I say it must be  abolished.   Try it by yourself.. it is inevitable!  Do you concur?
regards,     Dr. Mohammed Barzaq

Answer:

The foci-adjusted mean daily motion for the Earth, from a celestial mechanics table, is 0.9856874 deg/day. When this is multiplied by the correct time interval for each given “quarter” orbit the same area will be obtained. There is no "violation" of the 2nd law.

With sufficient accuracy each area mapped out in accord with the 2nd law is given by:

½  r ²  (D  Θ)

Where Θ is the angular difference between t2 and t1 in the orbit..

The rate of areal description (mapping)  is then usually divided by (D t) so:

A =   ½  r ²  (D  Θ)    / (D t)

But, since this rate is constant (by the 2nd law) we may write:

h  =   r ²   d ( Θ) / dt

where h, a constant is 2x the rate of mapping of area by the radius vector.  We may also express it in terms of three constants c1, c2 and c3 defining the orientation of the orbital place, e.g.

h =  [c1 ² +  c2 ²   +  c3 ²]  ^½)   

 Thus, n, the mean daily motion is the mean value of  d ( Θ) / dt

 For all points in the orbit.

Re: your specific examples of time intervals, i.e. March 21 to Sept. 21, and Sept. 21 to Dec. 21, it is evident to me that your construal of “unequal” areas is based on the misconception that the dates of solstices and equinoxes are absolutely  fixed when they are not.

 For example, the date for the winter solstice which you fix as Dec. 21 can actually be any of the dates: Dec. 20, 21, 22 or 23. That is as much as a 3 day spread which will make a significant difference in your computations.

The same applies to your dates from March 21 (vernal equinox)  – which can also vary, and June 21 (summer solstice) , ditto. My point is that when the proper specific dates are used (for the given year – they change year to year) you will find the same areal ‘map out’ for each quarterly interval of the orbit – even with the differing distances factored in.

Thus, it follows that when the greater distance (radius vector) is entered for one interval, the comparison interval (in days, and hence degrees per day) will be counter -balanced by a different date, e.g. for the winter solstice – which would have to be larger (i.e. Dec. 23 >   Dec. 21), to compensate for the lesser r.

Again, the error is in presuming fixed dates to mark the termination and initiation points of your orbital (seasonal)   intervals.

Holman Jenkins Harps On Dem "Lawfare" - But Will He Now Admit Trump Is Putin's Pet Monkey To Fracture NATO?

 

                                                              

         "Patience, my little tweety. Soon we both will rule"

"If Trump isn’t an active Russian agent, I’m not sure what he would be doing differently if he was."-  Washington Post comment

"Trying to imagine the unimaginable is a useful mental calisthenic. So, suppose Vladimir Putin faced this choice: He could assuage his fury about the Soviet Union’s disintegration by conquering Ukraine. Or he could destroy the cause of that collapse — NATO. Now, imagine that he might not need to choose, because of the American president’s obsession with seizing a possession of Denmark."-  George F. Will, 'Greenland, Minnesota - Another Day Another Emergency', Washington Post

"The bottom line is that Congress, SCOTUS, business leaders and a lot of the press (yes, clicks are good for business) are allowing or enabling a dangerous malignant narcissist to run the show the way he does. They should all be deeply ashamed." - NY Times  comment

"We elected a criminal and we are responsible for his criminal behavior at home and abroad. We should have known better. Now we are a great country in a serious period of decline and we cannot dump our failure on our troops. We need to make sure our Congress steps out of the shadows and demands that our president follow the rule of law. If he can’t or won’t do that then he needs to be removed, lawfully, as our Constitution provides." -  Washington Post comment

"Little" Holman Jenkins Jr. (he whose bellowing twaddle dwarfs his actual size), again can't help himself as he bellows in his latest screed (Now Lawfare Engulfs the Fed) that Trump has "justifiable" excuses for being the way he is. "The lawfare can of worms - its opening both repels and entices."

 He opens the 'can' thusly: 

"The ironies of Donald Trump are never- ending, because Mr. Trump is by nature a reacher and overreacher. Certainly no next president is likely to emulate the norm defilements Mr. Trump is guilty of.  Certainly not as likely as Trump himself was to seize on and emulate the norm-breaking of his Democratic enemies."

But, of course, this is arrant twaddle because what Jenkins Jr. terms Dem norm- breaking was in fact holding a deranged demagogue and 'mob boss' president to account. A guy who believed he could flout not just norms but commit every high crime, from shaking down leaders (like Zelinskyy in Ukraine), to inciting an insurrection (a real one!) on January 6, 2021, after spewing endless lies on how Joe Biden "stole" the 2020 election. To now threatening to seize Greenland by force. 

But Jenkins Jr's axe to grind goes back even farther to the Mueller investigation of the Russian influence on his 2016 election, rising to the level of conspiracy.  There were literally troves of substantiated material in Mueller's (2019) Report:. 

www.justice.gov/storage/report.pdf

 Including his indictments of 12 Russian (GRU) agents.

[Read the indictment here.]

Therein Mueller  documented over 272 direct contacts - including 40 in person meetings- between Trump campaign team members and Russian-linked individuals including GRU agents.  Further, at least 33 high -ranking campaign officials and Trump advisers were fully aware of contacts with Russia-linked operatives during the campaign and transition. This included Trump himself.  None of those contacts were reported to the proper authorities. 

Then AG William Barr shrunk Mueller's findings to a 4-page memo basically saying "nothing to see here" which was the cue for the pro-Trumpists to squawk ""Hoax!".

But if they had half a brain they ought to have asked IF this was so, why was former Air Force officer Reality Winner incarcerated for 5-plus years under the 1917 Espionage Act for using the internet to leak documents pertaining to Russian interference in the 2016 United States elections.  And this was done in Trump's first term meaning Barr's DOJ already had recognized a factual basis to prosecute Ms.Winner.  

That Trump's ties to the Russians extend back decades was already well known even before the release of the Mueller Investigation Report. As per an extensive piece in The New Republic (Aug./Sept. 2017, . p. 29):

"A review of the public record reveals a clear and disturbing pattern: Trump owes much of his business success, and by extension his presidency, to a flow of highly suspicious money from Russia. Over the past three decades, at least 13 people with known or alleged links to Russian mobsters or oligarchs have owned, lived in, or even run criminal activities out of Trump Tower and other Trump properties. Many used his apartments and casinos to launder untold millions in dirty money .....Taken together, the flow of money from Russia provided Trump with a crucial infusion of financing that helped rescue his empire from ruin, burnish his image, and launch his career in television and politics....It's entirely possible that Trump was never more than a convenient patsy for Russian oligarchs and mobsters."

Flash forward and it's evident past is prologue to the future of Trump's (and Putin's)  plans. He even thinks he has a basis for grabbing Greenland by force a la Putin with Ukraine. Hell, as per a Newsweek account yesterday evening, the Kremlin has even said "Trump will go down in history" if he snatches that territory owned by Denmark. See:

Greenland live: Russia says Trump will go down in history

One needn't be a member of Mensa to grasp that Trump wants Greenland not out of any need for "national security" but because Putin knows that a military invasion of Greenland by the United States would mark the end of NATO.   This has been Putin's wet dream since day one, getting Trump elected back in 2016. Knowing that Trump would be the most useful pawn or agent to finally demolish NATO. What better pretext that going after another NATO nation's territory?

 The United States already has an agreement with Greenland that gives us full access to the island nation for military bases, so why does Trump need to possess it? Because of Putin. It all goes back to Putin and Trump's "hours long" phone calls.  The 'master' roars his orders and his doggie jumps to obey. We've seen it time and time again and only the most obtuse or ignorant refuse to see through the jabber.

Even bonehead Holman's own WSJ editors see the prologue, as per their weekend editorial (The Greenland War of 2026 - Trump’s lesson in how to turn U.S. allies into China’s friends). Hence are more aware than Jenkins Jr. of the degree to which Trump is a real Russian apparatchik, noting:

"For more than 75 years, the fondest dream of Russian strategy has been to divide Western Europe from the U.S. and break the NATO alliance. That is now a possibility as President Trump presses his campaign to capture Greenland no matter what the locals or its Denmark owner thinks.

"Mr. Trump on Saturday threatened to impose a 10% tariff starting Feb. 1 on a handful of European countries that have opposed his attempt to obtain U.S. sovereignty over Greenland. The tariff would jump to 25% on June 1. Presumably this tariff would come on top of the rates Mr. Trump already negotiated in trade deals last year (10% for Britain, 15% for the European Union)."

The question is why Jenkins Jr. doesn't get it? Why is he so far up Trump's orange rump that he can't see the light available to everyone else with an IQ over 100?

As a weekend NYTimes piece (Trump’s Greenland threats send a visceral shock through Europe)noted:

"For the past year, Europeans have watched with alarm as Trump has appeared to try to actively tip the scales of the Ukraine war in favor of Russia. They accepted a humiliatingly one-sided trade deal last summer in the hope that doing so might buy leverage with Trump in Ukraine peace talks. They read with alarm a new National Security Strategy that was far more critical of European countries than of Russia and China, and which contained an explicit threat to intervene in domestic politics on behalf of parties that many Europeans consider extremist....Trump’s attempts to annex Greenland threaten to be the greatest geopolitical shock that Europe has faced since the Berlin Wall fell in 1989. That was a moment for joyous celebration. The only people celebrating this time will be Russia’s Vladimir Putin and China’s Xi Jinping."

The only rational and logical explanation that explains Trump's actions is that he's a Russian pawn or agent, doing Putin's bidding. Likely because the Russian has a 'pee tape' on him, as first exposed in the Steele Dossier.  See e.g.

James Carville: There Is A Pee Tape 

Steele Dossier a fake? Think again! As reported in the comprehensive 'Moscow Project":  Dossier - The Moscow Project

"Donald Trump’s allies in Congress and in the media have long attempted to use the Steele Dossier to discredit the Russia investigation. It has also been the subject of lawsuits filed by parties named in the document.

However, Steele’s main allegation has proven not just true but prophetic: Months before the FBI even acknowledged the existence of an investigation into Trump’s ties to Russia—even before WikiLeaks began publishing emails stolen from Trump’s opponents by Russian hackers—Steele had numerous sources both within and outside the Russian government confirming the Trump campaign’s collusion with Russia.."

The Financial Times in its Feb. 16, 2018 issue, validated Christopher Steele's bona fides as a trusted MI6 agent, noting he was "the UK intelligence expert on Russia".  In addition, citing James Nixey, the head of Chatham House's Russia and Eurasia program, confirming that key sections of the dossier read exactly as reports from the secret services". .  The dossier itself was actually several memos, based on conversations with Russian sources, that were written between June and December of 2016.

Never mind. Jenkins, ignorant as ever, babbles near the end of his horse manure:

"Mr. Putin knew before Americans did that the dossier was a fraud."

In fact, Putin knew it was totally accurate especially the section on the origins of the 'pee tape', because "Mr. Putin" organized it as a future kompromat instrument to bend Trump to his wishes. And for evidence just look at the extent to which he has succeeded! Even setting up a special paramilitary 'police' force like Putin has in Russia, while jettisoning all normal relations with Europe making it more vulnerable to Russian incursions. 

The recent WSJ piece ('Is The West Over?', Dec. 27-28, p. C1) cleared up the issue of the betrayal of the U.S toward Europe when it was evident from months ago. This after Trump fairly gushed over Putin in his red carpet treatment in their Anchorage meet. From then, serious European statesmen who had built an alliance based on thwarting Russian expansion now realized the jig was up. Traitor Trump was in Putin's pocket and there'd be no turning back. After all Trump had broadcast months earlier how enamored he was of snatching nations like Putin, threatening to grab Greenland and its resources from Denmark.

As the WSJ piece ('Europe Confronts Rupture', p. A8, today) notes, avoiding confrontation with the Trump mutation of the U.S. "is no longer working". We are now an enemy state thanks to Traitor Trump and in the words of one expert (Martin Jacob):  "There is the sense that now Trump needs to be confronted or - like Hitler- he won't stop. What's next, Iceland?"

And Trump's bestie Putin would love such a confrontation. After all:

"Even more important to Putin than all of Ukraine is the longstanding Russian objective to divide the trans-Atlantic alliance. So recent Trump and U.S. actions that break the cohesion and trust within NATO are a gift to Putin"  According to Doug Lute, a former U.S. Ambassador to NATO."

And what better person to bestow that 'gift' than Putin's U.S. pawn and foremost Russian helper, Trump?

So why then do the perpetual WSJ political hacks like Holman Jenkins Jr. continue to savagely attack or instantly dismiss Uber MI6 spy Christopher Steele and the Steele Dossier - when all are as real as a heart attack? Trump's current actions prove it!

 The reason is obvious: to try to rehabilitate Trump's image (as a damned traitor) before the mid-terms while distracting from the  fact this fucker - actually a convicted felon - has violated the Constitution more times than any other president in history.  In addition, openly pandering to and serving the Russian dictator Putin, even emulating him. UK Guardian writer Julian Borger has correctly pegged the U.S. invasion and overthrow of Maduro as the "Putinization" of U.S. Foreign Policy, e.g.

The ‘Putinization’ of US foreign policy has arrived in Venezuela | Venezuela | The Guardian

 The same 'Putinization' is what has Greenland and Europe on edge, believing Trump might act like Hitler did in 1939 with Poland and try to seize it-  repeating his use of force in Venezuela.

The real irony's intention - like Trump always invoking "the weave"- is to misdirect and confuse the media. Who are largely unable to see that  Trump is not a normal U.S. president but a foreign implant.  He has succeeded by effectively weaponizing language in the service of Putin to divide the nation. Also, his threats remain the primary ongoing engine for "the mob to quickly spiral into a desire for and celebration of conflict - as a means of collective invigoration and purification."  

All of this accounts for the  right’s descent into this abyss and GOP complicity with it.

 It can't possibly just end in 2026, or in 2028, or any other year because Trump voters' "rage quotient" - and stupidity will never subside. Not when one reads, as in the weekend Wall Street Journal, that "59% of blue collar workers remain committed to Trump."  So that will these buttbrains think when their medical costs go through the roof thanks to their pal, Dotard? Blame it on the Dems?

 Ditto for Trump's invasion of U.S. cities using his paramilitary masked goons. This will have the same exact effect, i.e. of energizing his MAGA mob into renewed celebration of conflict and acts of terror.   As Peggy Noonan pointed out in her relatively anodyne weekend WSJ column ('What's New In Trump Two?'):

"Something many Trump supporters won't say: They enjoy the suffering they've caused and not only because they're in charge of the ship now. But because many of Trump's opponents - who dealt out mortification- were comparatively affluent and accomplished.  What Trump supporters felt toward them was social and professional envy. Trumpism gave this flaw a new carapace of meaning: a political rationale that lifted it out of pure and eternal human spite."

Wow, Pegs, thanks for making it so clear. So these losers and dolts had a 'refined' reason to enable a Russkie plant to debase and defile our own nation as he destroyed democracy while helping his Russian masters take control after shattering NATO. With him as their useful idiot and underling, of course.

 This is the way weaponization of language via lies works, but Holman doesn't grasp it - perhaps because he himself engages in it. Look no further than his final paragraph:

"You only have to look at who's in the White House now to conclude the biggest beneficiary of these lawfare abuses (like the Russia investigation), turned out to be Mr. Trump himself. Restored to office by the foolishness and corruption of his enemies."

Actually, no.  Restored to office by the endless lies, corruption and foolishness of his Reich wing defenders in the reactionary media - like FOX and the Op-ed pages of the WSJ.  Columnists like Holman Jenkins Jr. who used their column space to spray caca & disinfo slop on the gullible brains of their readers. 

The only way it ends is if the lies end, and the only way that happens - so far as any rational person can see - is if Trump is no longer around to spew them so twits and trolls like Jenkins can circulate them. Holman's final test as any kind of journalist is to finally admit Trump is Putin's deranged pet monkey with which he will fracture the NATO alliance. By threatening to grab Greenland - like he has Venezuela.

See Also:

opinion content. America’s barbarians-inside-the-gates turn

Excerpt:

Historical sensibility tells us it is the barbarians who storm the gates. In today’s America, it is the other way round. Inside the citadel, the hordes are incinerating America’s traditions of law, civility and restraint. The civic-minded cry in the wilderness. Measured by the old era’s conventions, 

US President Donald Trump’s bonfire is only a quarter of the way through. Like so much else — the US Institute of Peace, the Kennedy Center, the Versailles-style White House ballroom, other people’s Nobel Prizes — Trump is rebranding the US as his own. As America prepares to commemorate its 250th anniversary, the republic is flirting with its own funeral.

And:

Russia Cheers the Growing NATO Rift Over Greenland

Excerpt:

Russian President Vladimir Putin has sought to undermine NATO for nearly two decades. Now, as President Trump pushes to control Greenland, Moscow is cheering from the sidelines.

Kremlin spokesman Dmitry Peskov appealed to Trump’s ego this week as the president pressed his pursuit of the Arctic island. “By resolving the issue of Greenland’s annexation, Trump will undoubtedly go down in the history books. And not only in the history of the United States, but in world history,” he said.

And:

With Threats to Greenland, Trump Sets America on the Road to Conquest - The New York Times

Excerpt:

It seems safe to assume that when Harry Truman forged NATO at the dawn of the Cold War, he never imagined that over the course of nearly eight decades the only country that would wage economic war and threaten actual war against the allies for the purpose of territorial conquest would be the United States itself.

And yet that is the reality of this upside-down, might-makes-right world of President Trump’s creation as he slaps tariffs on America’s treaty partners and holds out the possibility of using military force to strong-arm Denmark and its European friends into giving up Greenland, a territory whose citizens do not want to become part of the United States.

Never in the past century has America gone forth to seize other countries’ land and subjugate its citizens against their will. ..

Coercing a loyal ally into giving up territory over its adamant objections would have been seen not long ago as preposterous, even mad — indeed, one of Mr. Trump’s own cabinet secretaries in his first term privately considered it delusional when he raised it back then. But it is a measure of how much Mr. Trump has changed the definition of normal that his appetite for seizing land that does not belong to him is debated as a serious proposition rather than dismissed out of hand as a brazen violation of U.S. treaty obligations and international law.

And:

Donald Trump | Greenland

For Trump, Greenland Is About Legacy — and Humiliation

by Heather Digby Parton | January 19, 2026 - 6:39am | permalink

— from Salon

A few months after the 2020 election, New York Times reporter Peter Baker and his wife, the New Yorker’s Susan Glasser, traveled down to Mar-a-Lago to interview Donald Trump for their book “The Divider.” As Glasser wrote in the New Yorker on Jan. 8, they asked him in passing about his odd desire to take over Greenland, revelations of which had briefly appeared in the press and which they’d also heard about from some of his former staff. Trump told them he’d looked at the map and wondered, “Why don’t we have that?… Look at the size of this, it’s massive, and that should be part of the United States. It’s not different from a real-estate deal. It’s just a little bit larger, to put it mildly.”

It’s been speculated, notably by MSNOW’s Chris Hayes> that Trump was looking at the Mercator Projection map that we probably all remember from our grade school geography textbooks. For a variety of technical reasons, this navigation map distorts the size of the land masses near the poles. But it’s possible that Trump doesn’t know that and instead thinks that Greenland is about the size of the African continent. Greenland is about 25% bigger than Alaska, but it isn’t that big.

» article continues...

And:

European Union

Memo to Europe: Remember Neville Chamberlain

by Robert Reich | January 21, 2026 - 6:45am | permalink

— from Robert Reich's Substack

To: European leaders

From: Robert Reich

It is impossible to appease a tyrant.

You struck a trade deal with Trump last year. He is now threatening to rip it up and apply economic coercion and even military force if you do not allow him to annex Greenland. He is also on the brink of allowing Russia to annex part of Ukraine.

Today, on his “Truth Social,” Trump reposted a comment saying, “China and Russia are the boogeymen when the real threat is the U.N., NATO and [Islam].”

» article continues...

And:

• Anchorage Meet Proved Trump Is Still Putin's Bitch...

And:

It's An Effing Abomination For A Felon & Traitor - Who Pardoned Over 1,000 Insurrectionists - To Declare A "Crime Emergency" In D.C.

And:

Trump Deploys National Guard for Local Crime After Calling Jan. 6 Rioters ‘Very Special’ - The New York Times

And:

Donald Trump

The Strongman’s Test Run, and Why the Next Deployment Will Be Bigger, Longer and Deadlier

by Thom Hartmann | August 14, 2025 - 5:24am | permalink

— from The Hartmann Report

He says he’s deploying the military to Washington DC because of a “crime emergency,” but armies don’t do policing: Their job, and their training, is to blow things up and kill people.

They have no training in evidence-chain-of-custody, arrest procedures, civil rights protections, criminal investigation, or any other aspect of policing. Sending a militia to do policing is like inviting the neighborhood butcher to perform your brain surgery.

In America, it’s also illegal. Under Posse Comitatus, the American military is explicitly forbidden from engaging in any police activities against civilian populations. Even though the Trump administration is bragging that the National Guard arrested almost 50 people yesterday in DC, the Posse Comitatus Act consists of just one sentence:

» article continues...