Wednesday, September 30, 2020

Trump The Racist Thug Proves He's Unfit To Debate - Or To Lead This Nation


                     "I want attention! And I'm gonna interrupt to get it!"

                       "Proud Boys" ready to stand by as per Trump's order.

"We intend to stop Antifa  raiding our cities and burning them down"

"An absolute disgrace!'

"It was like an abused mom trying to control a psychotic 8-year old!"  

 "We've just witnessed and heard the most monstrous cavalcade of lies ever spewed in debate history!"

"We just beheld Trump's plan in the debates: to ignore them and just flout every norm and rule to talk whenever he wants, say whatever he wants."

"I don't think Joe Biden is on the ballot but we are as a nation. We have a very clear choice in five weeks to decide what we love in common. Do we love decency, do we love democracy, do we love the rule of law. And if we don't do we just head off tht cliff totally..."   - Historian John Meacham on Today, this morning 

"Trump  put on a performance so contemptuous, so puerile, so dishonest and so across-the-board repellent that the moderator, Chris Wallace, morphed into some amalgam of elementary-school principal, child psychologist, traffic cop and roadkill." - Frank Bruni, NY Times this a.m

The preceding remarks were just some of the reactions that followed the 90 minute farce that unfolded last night, a disgrace and national embarrassment in every sense of the word.  As Janice said, "I hope to hell no one in other parts of the world watched this Trump freak show!'  But they probably did, we are still waiting to hear from family in Barbados.  

 All this debate did is confirm for me that I was right all along in asserting in my August 4th post, that it  would be a total waste of time for Biden, adding:  "Biden shouldn't dignify this mad monkey with a gun via formal debate."   Which is how I saw the unseemly cage match last night with moderator Chris Wallace looking more like an abused spouse, constantly trying to placate and appease the rabid ape who refused to control himself. (I.e. "I think you're gonna like the next question, Mr. President.") 

But this is how it went: Trump and Joe Biden sparred bitterly during the first presidential debate of the general election with Trump trying to bully Biden into submission and silence.  He hurled on  personal insult after another  as the pair clashed over healthcare, the coronavirus,  the economy and the Supreme Court.

Ignoring the rules, Trump repeatedly interrupted Biden. a total of 741 times, according to the online tracker Factbase feed. The thug's interruptions were often in the middle of Biden's 2 minute statement-   which had analyst Chuck Todd asserting in the aftermath it was Trump's intent to sow chaos to try to score a win.   Biden, amidst this barrage of bullying and bombast, kept his cool, didn't blow his top.  But Joe did allow himself righteous anger when - at the end of his tether from Trump's histrionics- he told the bombastic buffoon: “Will you shut up, man? This is so unpresidential.

As  Frank Bruni wrote in his column today:

"To the extent that Biden occasionally flung mud of his own, well, when you’re dragged into the pigsty, you have no other choice."

I totally concur. Some media outlets tut-tutted at this   ("Biden resorted to  too many pejoratives!") but as Claire McCaskill said on the NBC post-debate analysis: "Sheesh! If it was me I'd have throttled him!"   So would most sentient people.    As I told Janice, I'd have grabbed a coke bottle and broken it over his skull, had I been interrupted in a debate like Biden was.  So Biden's mild displays of justified anger, including calling Trump a racist, clown and liar, were tepid by comparison - especially given how much time Trump stole while Biden was trying to make his points.

At another point, Trump even wrestled with the moderator, Chris Wallace, complaining: “I guess I’m debating you, not him. But that’s OK, I’m not surprised.”  At that point Wallace ought to have slapped the orange Orang down and made it clear there were ground rules and Trump wasn't following them. (But then as Maddow noted in the MSNBC analysis, this debate was in fact a microcosm of Trump's flouting of rules and norms of government.  And to the point, he "acted like he didn't even want to be there or want an election.")

By the end of the first hour I had tallied 133 lies spouted by the orange cockroach. Around that point Biden aptly said:

The fact is that everything he’s said so far is simply a lie. I’m not here to call out his lies. Everybody knows he’s a liar.”

Of course,  we had reason to worry this debate would degenerate into the side show it became when we learned of the Trump team's pre-debate shenanigans.  For example, Trump’s communications director, Tim Murtaugh, alleged that Biden’s campaign had reneged on an agreement to a “pre-debate inspection for electronic earpieces” and that his team sought “multiple breaks during the debate, which President Trump doesn’t need”.   The first is based on a long running conspiracy theory on the Right that Dem debaters needed hidden earpieces. Of course this is utter horse shit. The Right's minions and a-holes cling to this because they know THEY can't keep two consistent facts in their head for 90 minutes so ascribe that weakness to opponents. Hence the need for hidden earpieces they presumed.   The second claim, about Biden demanding "multiple breaks" is simply twaddle.   A lame effort to cast aspersions on Biden's age and suggest a weak bladder.

Then there was Trump’s demand that Biden submit to a drug test, a suggestion Biden laughed off,  as he should have.  It was one more indication of Trump being a "nutjob"  - a term used by Janice's cousin Geoff'- after he weighed in.  (Also suggesting the need for Trump to be escorted to a "rubber room" for ECT before any further debates). 

The debate was arguably Trump’s best opportunity to reset the presidential race in which he is currently behind.  But both the NBC and MSNBC teams of analysts believed he only barely kept his own supporters in place.  His performance "definitely did not win any suburban, college educated women" - in the words of Nicole Wallace. 

The economy is the sole issue where Trump retains a slight advantage over Biden, but that also should soon be history as the truth of Trump's poor economic stewardship slowly dawns, and the facts become more evident. Including that Trump's early economic gains - including adding jobs- were largely a result of the benefits accrued from the preceding Obama administration.  All that's happened under Trump is mammoth job losses, and monstrous  compounding debt - now nearly $4.8 b a day, as blogger Robert Freeman has noted, e.g.

As for "undecided" voters, all they need to know - to take away - from this debacle was that Trump refused to condemn his white supremacist supporters. In response to Biden challenging him to condemn his far right supporters, Trump simply directed his comment to the renegade Proud Boys, telling them to "Stand back and stand by".  Adding: "Somebody’s got to do something about antifa and the left.”  MSNBC analysts showed Trump's initial words "stand back, stand by". had within minutes appeared on the group's logo.

THIS is the swine we have defiling the office of the presidency, a racist at heart, and why we need to get rid of his sorry ass come November.   As one WaPo columnist correctly summed it: 

 "It was frustrating to hear Wallace calling the president “sir” as he pleaded with him to adhere to the rules to which he had agreed. Sir. Trump did not deserve that nicety because he did not come to the debate bearing the mantle of the presidency. He came with the demeanor of a thug."

Frank Bruni, in his Times column, has said Biden should engage in no more debates given Trump's reprehensible behavior, emulating a deranged and rabid ape.  I would agree but leave only one option for the next Town Hall debate: have a controller pull the plug on Trump's mic if he interrupts or talks over.  There must be control and following the (agreed upon)  rules else we have nothing but chaos, also a gross disservice to the electorate.

See Also:

Acrimonious debate sparks calls for new rules to rein in Trump and questions about format’s usefulness


Trump refused to condemn white supremacists, militia members in debate marked by disputes over race


Tuesday, September 29, 2020

All Biden Needs To Do Is Keep His Composure (when Dotard mounts personal attacks) And He Will Win This Debate


As I  wrote in my August 4 post, I see no reason that Joe Biden ought to agree to debate the 74 year old narcissist brat in the White House, with pretensions to absolute power. That  assessment doesn't change now, the day of the debate,  given we've beheld Trump's mental condition has further deteriorated.   Also his lies and gaslighting have grown to malignant proportions.  Look no further than his continued attacks on mail voting, as well as admitting on tape (with Bob Woodward for his book, RAGE) how he lied to downplay the virus- allowing tens of thousands to die from Covid.  

 Thus, we've already got prima facie evidence that Trump has failed the essential test for presidential leadership by failing to protect the citizens under his "mantle" of leadership.  We don't need any more artificial tests like a staged debate, and the only point in having one would be to disclose or confirm (for the "undecided" voters) what is already known by the more educated and well-read voters: i.e. that Trump is a mendacious and unstable cretin who also poses a security risk to this election.  That includes the hundreds of millions he owes to unknown lenders, which will come due during a 2nd term.  The American people deserve to know WHO these creditors are. Maybe Russian Oligarchs?   (Pay attention, Joe, in case Trump comes after you with a barb attacking Hunter, or other family members.)

 As  Stephanie Ruhle said this morning:  "We can expect Biden will go there even if moderator Chris Wallace doesn't."   Well, we would hope so!  Biden's byword this evening needs to be:   Don't let sleeping counterattacks lie around.

Minus that, there is nothing at all to be gained by citizens from witnessing tonight's event.   Moderator Chris Wallace has vowed to make himself "invisible" so that means he likely won't intervene when needed, including fact checking.  And we know all that Trump will deliver is lies, and those mixed with bombast and name -calling, so what's the point? There is no educational aspect whatsoever.  As journalist Elizabeth Drew - a panelist in one 1976 debate- put it:

"The debates have never made sense as a test for presidential leadership."

 John Sides wrote in Washington Monthly magazine in 2012:

That presidential debates can be ‘game changers’ is a belief almost universally held by political pundits and strategists. Political scientists, however, aren’t so sure. Indeed, scholars who have looked most carefully at the data have found that, when it comes to shifting enough votes to decide the outcome of the election, presidential debates have rarely, if ever, mattered.”

More to the point, there are likely barely 3 percent of voters who can be influenced at this time.  Three percent one way or other can still make a difference so there is some small merit in Biden  vigorously competing to win - say as opposed to "phoning it in" - with a merely acceptable performance.

How best to do that is easy:  Biden simply needs to  avoid blowing up, losing his cool, in front of a projected audience of 50 million.  If Trump attacks at a personal level, Biden has any number of low hanging Dotard "fruits" to go after in brisk ripostes, including:

- Calling attention to how big a loser Trump is, i.e. more successful at playing a billionaire than  being  one.  

- Calling attention to his massive personal debt of $300-400 million (as the Times'  report disclosed) and pushing Trump's buttons as to whom this debt is owed. Russian oligarchs? Mafia?  Emphasize the national security aspect, and get Trump's buttons properly pushed so HE explodes on stage, rope a dope if you will.

 -  Remind voters of how this swine called fallen military "losers" and "suckers", like when he went to the cemetery near Belleau Woods.

-  Remind voters- viewers of how he lied about the virus as exposed in Woodward's tapes for his book, RAGE. Deliberately downplaying it even when he acknowledged it being much worse than seasonal flu.  Then ask him directly: "How could you do that and let tens of thousands more Americans die because of your utter mendacity?"  - Then watch his eyes bulge as  he blows up.

All of the above can be delivered at the right time in rejoinder form, and without any need to scream or even raise one's voice,  or lose one's composure.  Simply deliver the verbal jab with a cool smile or nod of the head,  like a shiv to the spine of a  known predator.   Any of these attack topics  can be interjected after Trump  launches a personal attack, or if  Chris Wallace is not inclined to bring up these recent Trump atrocities.   Yes, the chosen topic for attack (like his massive debt) may not be on the debate list but that is no reason to let it sit unused, especially if Trump earns it with a personal attack.

Biden ought to also prep a good humorous response to an issue or Trumpism,  and look for a timing opportunity to use it.  E.g. if for the nth time he blurts out: "Fake News!  Fake News!"

"There you go again, Mr. President!  Is it that difficult to get your brain off that single track? Maybe we need a retest of your mental ability."

After all, authoritarians hate nothing more than having their supercilious, bloated egos punctured by a quick jokey takedown. 

The main goal for tonight: Get the ever incandescent idiot Trump to blow his lid on national TV - while you simply smile as he drags himself and his campaign  down to greater depths.

See Also:

What to know about the first Trump-Biden debate



Trump’s debts and foreign deals pose security risks, former intelligence officials say

Monday, September 28, 2020

Why I Can't Agree With Bob Woodward - Re: "Some" Of Trump's Followers Being "Decent People"


                        Some of Trump's 'very fine' people

Bob Woodward, author of  'RAGE',  wants us to bear in mind that: "There are some fine and decent people who support President Trump.  These include farmers, military people,  health care workers, policemen and others".   This - related at the end of an interview with Chris Hayes (on his ALL In show) - some 2 weeks ago, after Hayes asked if Woodward judged any of Trump's supporters.

Sorry, Bob, but I'm not buying it.  "Fine" and "decent" people do not descend  on polling venues  - like in Fairfax, VA recently- and seek to disrupt early voting, e.g.

These are not "fine and decent" folks but brainwashed, angry goobers.  Worse,  even if you are not precisely among these miscreants - if you support Trump - you still end up enabling them via their mindless backing of Trump.  In the case of the preceding lot, they actually had the gall to chant "Four more years! Four more years!"  at voters went to the polls, like their impoverished brains had really taken leave.  

But then they inhabit an alternative universe and that is exactly the problem.  A universe where masking up equals bondage and slavery, and medical professionals who recommend masks, social distancing are traitors, and governors trying to save lives using lockdowns are threatened by extremist gun nuts toting AR-15s into state houses.  (Even calling for the beheading of Michigan Governor Grace Whitmer.)

This alternative universe of fantasy and crackpot conspiracy ideation has largely been created by Trump in recent years, via his endless stream of lies, misinformation, dog whistles and gaslighting.  Journalists now tally more than 20,000 lies since he assumed the presidency and also chronicle accusations of sexual misconduct against him (26 so far).  And yet he seems coated with Teflon, given nothing seems to stick.  This is because of way too many "fine, decent" people who refuse to see through the sham. As Janice put it: "Ignorance is no excuse especially after he had infants caged and separated from parents 3 years ago. Also, now admitting on tape he deliberately lied about the virus!"

 Why doesn't it "stick"?   Well,  because like all effective authoritarians he creates diversions and nonstop chaos with ever more outrageous claims and deranged statements almost every day.  Like his ongoing barrage against voting by mail,  including insisting he will not accept any peaceful transition. 

 In such a turbulent atmosphere, the normal mainstream media can't keep up.  Hell, they are still overly deferential to this walking turd.  One only had to watch his presentation of Amy Coney Barrett Saturday where all the press corps uniformly rose and clapped, like a bunch of trained seals.  They ought to have booed the bastard like those marking RBG's lying in state Thursday night when he appeared.

It is this chaotic atmosphere plus Trump's tidal wave of lies and gaslighting that have helped to create his legions of  zombies, and brainwashed faux "liberty" nincompoops. We cannot embrace this lot into any political comity until they return to reality, and there's no sign they will.  As Maureen Dowd wrote in her Sunday column:

"The cultural ecosystem, and the fever swamps of social media that amplify Trump’s craziness, will remain. Fox News and Facebook will continue to validate the biases and conspiracy theories of a nation that’s increasingly proud of its ignorance, anti-intellectualism and denial of science."

Times' columnist Frank Bruni basically seconded this outlook with his own take:

"We’re in terrible danger. Make no mistake. This country, already uncivil, is on the precipice of being ungovernable, because its institutions are being so profoundly degraded, because its partisanship is so all-consuming, and because Trump, who rode those trends to power, is now turbocharging them to drive America into the ground. The Republican Party won’t apply the brakes

Bruni pointed out that the riff raff and renegades trying to intimidate voters in Fairfax was "no rogue group" adding: "This was an omen and a harrowing one at that."  

 Indeed it is, because with every day that passes with Trump yapping about "fraud" and "mail ballots" his zombies and crazies get more and more worked up.  Add in  incendiary stunts like the one about "discarded Trump ballots" in Pennsylvania - and you are looking at tossing gasoline on an already lit fire. (In this case, it was later determined the allegedly "discarded ballots" were a result of Pennsylvania Republicans passing a law banning "naked' ballots, i.e. those not inserted into an outer security sleeve.  So when 9 such ballots were accidentally found they were discarded by bureaucrats.  There was no sinister element as claimed by Toady Barr, or Trump or their lackeys.).  

Then there is the pandemic and the University of Washington model now forecasting quadruple the daily deaths from Covid by year's end.  The source of  90 percent of the problem? Trump and his endless lying and minimizing the impacts while corrupting the science including at the CDC - all the time giving false hope of a vaccine- for political gain.  As medical specialist and journalist Laurie Garrett said in an interview on Saturday, it is "ridiculous" any viable vaccine could be ready even by year's end given intermediary trials haven't even begun for most candidates.    Meanwhile, Trump's zombies continue to attack health specialists, and defy mask orders - the most effective way to control the virus pre-vaccine.

Also derivative of pro-Trumpian insanity  (some might say the core of it) is the QAnon movement, which Bloomberg columnist Andreas Kutsch has recently described as the "vilest" conspiracy ideation since the medieval slanders spread against the Jews, e.g. poisoning drinking wells and drinking the blood of Christian children.  In the QAnon case, deep state "elites" (including prominent Dem politicos) are accused of child sacrifices to Satan and organizing pedophile rings. Hence, a major example of current mass insanity on the tribal conservative Right.

One Colorado Springs contact tracer - John Potterat- when interviewed by the local Indy paper three weeks ago, said:

"As I look at American society today we're simply not the people we used to be. I'm disappointed. The American population was much more society conscious 30, 40 or 50 years ago.  We're not the people we should be."

We're not the people we used to be primarily because we've been fractured into two hostile tribes on account of one tribe becoming detached from objective reality.  This is the tribe that's been brainwashed by Limbaugh, FOX news and that follows Trump's every word.   They intimidate blue state governors with loaded rifles at state houses, attack medical personnel on social media and at their homes, mock masks - and deride virus deaths and cases with  screams of "Hoax!".  Never mind Covid has now felled 204,000 fellow citizens.  They are also prepared, again, to try to commit "national suicide" (in the words of Keith Olbermann in his recent book 'Donald Trump Is F*cking Crazy!') and take the rest of us down with them by voting Trump back in.   Make nice or judge kindly those prepared to do this, because they're too damned lazy or dense to see the scam artist con man authoritarian trying to pull us down his sinkhole?  No way in hell.

My only real worry is how even at this stage Trump's Svengali-like mind control is working on likely voters as per the economy. Polling from NBC/WSJ as of Sunday showed Trump with a 53 %- 42 % edge over Biden as regards handling of the economy, which is absolutely preposterous. In fact it's insane because there's zero basis in reality for it- and that was before the NY Times bombshell expose yesterday showing this fruitcake may owe up to $421 million to unknown lenders (as MSNBC's Stephanie Ruhle noted this a.m. and which personal liability he'd carry into any new term.)  This is apart from his using twitchy con-man strategies to get away with paying only $750 in taxes (in 2016, 2017), when for other years he didn't pay a dime.

 Leaving out his tax cons, e.g. taking tax deductions on his personal expenses - like over a million for "housing" and $70,000 for "hair styling and hair spray", we now know how his policies have hobbled the U.S. in terms of borrowing and debt - as Robert Freeman (2nd link below) notes:

"This borrowing masks profound weakness in Trump’s “best economy in the history of the world.” It amounts to $4.8 billion of borrowing every DAY since Trump took office. Yes, injecting almost $5 billion of borrowed money into the economy every day for almost 4 years can create the illusion of prosperity, at least for a while. But borrowing is not a sign of economic strength, but of weakness."

 Beyond that, we know over 28 million jobs have been lost and we are in the midst of a historic recession all tied to the pandemic - which management Trump has blown, fucked up to a far thee well. Further, annualized growth under Trump ranked seventh among his 11 predecessors. And growth actually slowed during each of Trump’s three years.   Handle the economy better than Biden?  Only a brain-eaten imbecile would believe so.

So why is so much credit being given to Trump for his handling of the economy, which is tied to the pandemic which now has seen over 204,000 dead?  Have the respondents' brains been taken over by brain-eating amoebas? It appears so. 

Apart from the apparent mental regression spreading into the larger (voter) population there is the danger of overt voter suppression. We now know - thanks to reporting by Michelle Goldberg- that Republicans are planning to have tens of thousands of volunteers fan out to voting places in key states, ostensibly to guard against fraud but effectively to create a climate of menace. Trump has not merely blessed such renegade tactics but encouraged them. On Fox News last month, he bragged to Sean Hannity about all the “sheriffs” and “law enforcement” who would monitor the polls on his behalf. At a rally in North Carolina, he told supporters: “Be poll watchers when you go there. Watch all the thieving and stealing and robbing they do.”

Michelle Goldberg does, however, believe there's a way out of Trump's 'through the looking glass' world of unreality, writing in a column from 3 weeks ago:  

"Trump's recordings (e.g. on the Woodward tapes) offer the prospect of breaking through his alternative reality, of nailing down this most slippery and mendacious of presidents by showing everyone who he really is."

But wisely adding:  "Our politics suffers no shortage of incontrovertible proof of Trump's venality. What it lacks is accountability"

A first start in reversing this could be made at the first presidential debate tomorrow night, if moderator Chris Wallace would play at least one of the Woodward tapes and demand that Trump respond.  Especially the one where he admits to deliberately playing down the virus despite knowing of its real fatality rate being worse than flu.

If Wallace and other debate moderators lack the nerve to 'take the bull by the horns' it is possible (according to Goldberg) that: "Woodward's utterly damning tapes of Trump discussing the coronavirus will fall into this same 'nothing matters' cycle."   And if that happens Trump scores another 'W' thanks to the lackadaisical media.

Shortly after Trump was elected, the Russian-born journalist Masha Gessen published an important essay called “Autocracy: Rules for Survival.” Gessen laid out six such rules, each incredibly prescient. This was notably after taking Obama, Hillary, and members of the "liberal" commentariat (like Nicholas Kristof and Thomas Friedman) to task for too quickly being willing  to say we ought to close ranks behind the orange maggot and give him a chance.  (At least Keith Olbermann never bought into this horse manure.)

Leaving out that brain fart and misfire at the start of Trump's reign (and note I never accepted him as a president, at least mine)  I most often hear one Gessen rule repeated, i.e. on 'ALL In': “Believe the autocrat,” which says:

 Whenever you find yourself thinking, or hear others claiming, that he is exaggerating, that is our innate tendency to reach for a rationalization.

That tendency is clearly one we must avoid as we near the home stretch of this most critical, most existential of elections. I mean, it isn't often that a suicide attempt - say by hanging -  survives. (In this case, as per Keith Olbermann, a national attempt).  But good god, to deliberately make a 2nd attempt and succeed?  Well it's beyond the pale, in fact beyond any dystopic ideation I can imagine short of a Torino 10 asteroid strike.  If it does happen to succeed this time and  democracy dies under a 2nd Trump term, we can thank his "fine and decent" followers who joined ranks with the Nazis, white nationalists,  KKK,  extremist militias, QAnon  conspiracy degenerates and others to allow Trump to complete the destruction of our nation he started with his first term.

Make common cause with them? No way in hell!

See Also:

Solutions To Partial Differential Equation Problems

 1)  Solve:     z/  x   =   ax +  y

SolutionSince the differentiation is with respect to x only, we can treat y as a constant and separate the variables thus:

dz =   (ax + y) dx

On integration we obtain the general solution:

z   =    1/2  a x 2     +   xy   +   y  (y) 

2) Solve:   2 u/  x2  -  4  [  2 u /  x   y]  + 4[  2 u/  y 2 ] = 0

Solution:  The auxiliary (quadratic) equation corresponding to the differential equation is:

m2  - 4 m  + 4 = 0

Factoring yields:   (m - 2)  (m - 2)  =  0  

This discloses a double (e.g. repeat) root of m  = 2

So the general solution is written in terms of two arbitrary functions, 
each with (y  + 2x):

Thus:   u   =   f( y + 2x)  +  xg(y + 2x) 

3)  Solve the following higher order partial DE:

 4 z/  x4  -   6  [  4 z/  3   y]  +  14 [ 4 z/  x2   y2 ] -  

 16 [  4 z/     y 3 ]     +   8  [  4 z/  y 4 ]   =  0

Solution:  Again we write out the auxiliary (quadratic) equation corresponding to the DE:

m4  - 6  m3 + 14 m2  - 16 m  + 8  = 0

Factoring yields:  

(m - 2) (m - 2) (m - (1 +i))(m - (1 - i))

so  m  =  2, 2,  1 + i, 1 - i    

Hence, the general solution (noting the repeat root of m = 2) is:

z  =   o (y  +   2 x)   +  x  y o (y +  2 x)  +  j1  (y  +  x  +  ix)   +   j1 (y   + x  - ix) 

+  i[ y1  (y   + x  +  ix)  -   y1  (y   + x  -  ix)]

3)  Consider a wire of length  ℓ :

]--------------    ---------------------->[

for which the relevant wave equation is:

  2 u/  t 2  =   c 2   [ 2 u/  x2 ]

If the displacement satisfies:

u(0, t) = u( , t )  =   0     ( t >  0)

Suppose at time t= 0 the displacement is u(x,0)  =  f(x)

And:     u/  t t=  0   =  g(x)

Using the technique of separation of variables write two different general solutions for X(  ) and state why one is unacceptable

Hints:  i)  u(0, t) =  X(0) T(t)   and

 (ii)  u( , t )  = X(  ) T(t)

Solution:  By separation of variables we have: 

u(x, t) =   X(x) T(t)

Now let:  dT/ dt =  T ',    d Tdt 2   = T"

dX/ dt =  X '        d 2X/  dt 2   = X "

So that:   2 u/  t 2  =  X(t) T "(t)


 2 u/  t 2  = X" (x) T (t)

The wave equation can then be written:

XT "   =   2   X"  T

Or:    T "2  T   =    X" / X

We set both sides equal to a constant K (separation constant):

 T "/ 2  T   = K  =      X" / X

Two equations result:

i) X"   -   K X = 0

ii) T "  -   2 KT    =   0

We know (hints):

a) u(0, t) =  X(0) T(t) =  0 

b)  u( , t )  = X(  ) T(t)  = 0 

Look at two cases:

i) K  = 0

Then X" - KX = 0  becomes X" = 0

For which: X(x) = c1 X + c2

And: X(0) = 0   è   c2 = 0  so X(x) = c1 X

However,  X(x) = 0 ⇒  c1 = 0  and X(  ) = 0

Which is unacceptable, so K cannot be zero. So next look at K > 0

ii)  K  > 0

Then:  Let K  =    r 2

So that:  X"(x) -  r 2 X  = 0

Has solution:  c1 exp (r x)  +  c2 exp (-r x) 

After eliminating c2  (as fn. of  c1)  

 X(x) =  c1 [exp ( r x)   -   exp (-r x) ]   

X() =  c1 [exp ( r )   -   exp (-r ) ]   =   2 c1 sin n r ℓ