Not quite a differential equation...

[@godfreypigott]

Who would have thought maths was an outdoors activity!

[@isaacaguilar5642]

Newton

[@tomatrix7525]

He is getting upgrades done to his setup, looks like Penn decided to bring the board outside

[@nikitakipriyanov7260]

you can find videos on Michael's channel where he is doing backflips on board erase cuts

[@l0remipsum991]

audio is quite good outdoors, that helps with the presentation

[@DeanCalhoun]

having a backyard blackboard is now a new life goal of mine

[@bobdowling6932]

Now that the videos are outdoors there’s nothing to stop you backflipping again.

[@BikeArea]

When did he execute a backflip? 👀

[@henrymarkson3758]

Michael Penn has definitely set the bar higher for Maths on KZbin. I would also like to (highly) recommend Michael's 94 video playlist on Differential Equations on this channel. It's gold.

[@alexwolffe7805]

Love how simple you make all these problems.

[@cycklist]

I love how Prof Penn is always innovating. That's why he is the best maths channel on KZbin.

[@nikitakipriyanov7260]

I knew this trick with operator factorization, this way Dirac equation is deduced from Klein-Gordon equation; however, I didn't expect this could be used to solve an inequality.

[@YoYo10542]

I love the backyard filming! There’s something so playful and whimsical about it.

[@goodplacetostop2973]

12:51 Outdoors video PogChamp

[@sinecurve9999]

Mathematics in nature! Love it!

[@joaovitorcb]

That was amazing! Please solve more from IMC

[@heathpearson6375]

It's quite straightforward to show this by solving the inhomogeneous equation Lf=u where u is a positive source term. You can find the Green's function of the IVP and then the solution is given by 3e^2x-2e^3x + an integral of the Green's function times the positive source term. You can show that this integral is positive and so the result is shown. Requires no sneaky identities or tricks.

[@edskev7696]

Do you mean Lf = u where u is a positive source term, and L is the differential operator?

[@heathpearson6375]

Ed Skev I mean to say that u is the function here and f is the source term. L is the differential operator. I didn’t use the same notation so what you said is what I mean

[@oskarjung6738]

Using green's function means using pseudo maths that involves the heaviside function which is not even a proper function. So, please keep your engineering/physics tricks to yourself.

[@positivegradient]

Both, the location and the problem, are super cool

[@txikitofandango]

Love the outdoor setting

[@akuaku77]

I like the backdrop much better than the usual.

[@stvp68]

Reminds me of when my college math class would meet outdoors

[@Tiqerboy]

Wow, thank God I didn't have to take the Differential Inequality course as an undergrad. The solution to that was quite complicated! I was able to remember how to solve the differential equation though :-)

[@andrewdsotomayor]

Good throwback to the older vids

[@OuroborosVengeance]

I loved this simple, easy to follow example. I'd love more videos like this

[@keksauraisks]

That's a really interesting method. I've never seen it before!

[@shahinjahanlu2199]

Me too

[@riadsouissi]

Solving the DE=a with a>=0 is also straightforward. We. Get f(x) = - 2e^3x +3e^2x +a*h(x) where h(x) = 1/6+e^3x/-e^2x/2 We can show h(x) is monotone because h'(x) >=0 and hence h(x)>=h(0)=0

[@nikolatesla6662]

JUST LOVE IT PENNN!!!

[@andcivitarese]

Algorithmically: rewrite the inequality as an Ordinary Differential Equation f''-5f'+6f=6k with k nonnegative. As shown by Michael the solution to the homogeneous differential equation is a linear combination of Exp(2x) and Exp(3x). A simple particular solution to the complete equation is the constant (and so with vanishing derivatives) f(x)=k (for this we introduced the factor "6"). So the general solution of the ODE is f(x)=AExp(2x)+BExp(3x)+k. Using the initial conditions you find A=-3(k-1) and B=2(k-1). Finally, putting all toghether and expanding you have f(x) = -3kExp(2x)+2kExp(3x)+k+3Exp(2x)-2Exp(3x). The last two terms are the given lower bound. You can factor the other terms as k(Exp(x)-1)^2(2Exp(x)+1), which is a nonnegative stuff for nonnegative k. GPS

[@kezza7773]

You cannot assume k is constant.

[@nikoladjuric9904]

It could be f"-5f'+6f=k*e^(qx),k>=0 or any nonnegative function (example: x²+x+1), so you solved one special case...

[@tahirimathscienceonlinetea4273]

It's very nice proof using the second method associated with nice tricks 👍👍👍

[@ratnakermehta7383]

Just Beautiful! And yes, "A Thing of Beauty is a Joy for Ever"!!

[@tomatrix7525]

These are top class. Please do some more differential eqns. If that is simethng u oike

[@reshmikuntichandra4535]

Michael, I suggest you to try some problems of the 2021 INMO (Indian National Mathematical Olympiad). It was sooooo hard.

[@martinnyberg9295]

I vote for more outdoor lectures! 😊👍🏼☀️

[@roberttelarket4934]

Dare you Mike to do a video for some solution while climbing vertically!

[@anggalol]

Nice background, i was expecting kindergarten bg lol 😂😂😂

[@ImaginaryMdA]

Oh that's lovely!

[@goodplacetostop2973]

Not a homework today but a suggestion (that I’ve submitted through the the form too). But I know some number theory addicts will be eager to solve it, so... There are decimal integers whose representation in some number base B = 2, 3, 4... consists of three nonzero digits whose cubes sum to the integer. For example, 43 (base 10) = 223 (base 4) = 2^3 + 2^3 + 3^3 134 (base 10) = 251 (base 7) = 2^3 + 5^3 + 1^3 433 (base 10) = 661 (base 8) = 6^3 + 6^3 + 1^3 Prove that infinitely many such integers exist. SOURCE : Crux Mathematicorum Vol. 5, No. 9 (November 1979). Submitted by Allan Wm. Johnson Jr., Washington DC.

[@pardeepgarg2640]

How you came so early , btw I am 2nd

[@pardeepgarg2640]

Solution: I don't know

[@srijanbhowmick9570]

Since I don't know the exact definition of a digit hence I am asking , does "digit" always mean 0,1,2,...,9 or in base x , "digit" means an integer between 0 and (x-1) inclusive ?

[@abhilashsaha4590]

@@srijanbhowmick9570 I think the second one.

[@srijanbhowmick9570]

@@abhilashsaha4590 I also think so because "decimal digits" is also a mathematical term so it must be a type of digits 👍

[@lucyhaddant1303]

Thanks a lot.

[@CraigNull]

This is a 2nd order homogeneous DE with constant coefficients and initial conditions defined at 0 in which you can change the "=" sign of both the original equation and of the solution to ">=" and the conclusion remains true for x >= 0. What's the largest (simply stated) class of DEs where this naive conversion to an inequality works?

[@shahinjahanlu2199]

Great job

[@elib2670]

Math teachers talk about the characteristic polynomial with an air of mystery. . . but it’s literally just guessing y=e^ux

[@hybmnzz2658]

Because there is apparently a linear algebra connection. I still have no idea if it has anything to do with the Wronskian matrix and it's characteristic polynomial.

[@mujiburrahmanhelal6459]

I like your videos always.keep it up sir.respect from Bangladesh

[@jingermcblabbersnitch7162]

"and that's a good place to"

[@MathElite]

Love the video I like the old setup better though nice work

[@MichaelPennMath]

The studio is undergoing renovations so I put this together in the mean time.

[@MathElite]

@@MichaelPennMath Ah ok cool

[@rockyminarro1715]

Could this problem be solved using Laplace transform directly from the inequality?

[@BikeArea]

The problem magically (dis)solves itself in rainy conditions. 😮😊

[@SlidellRobotics]

Next time, put a couple of holes in the bottom of your chalk board and put a line through them and around the trees, and your left hand can relax a bit!

[@utkarshsharma9563]

This is some kind of big coincidence, because a few days ago, I came to know about the IMC olympiad, and today I learnt how to solve the differential equation you were solving in the beginning 😁

[@prathmeshraut1616]

Out door Maths Never seen before

[@domc3743]

Was able to solve this using differential equations tricks I learnt recently learning harmonic damping. Felt a bit cheaty because I have no understand of how it works. It's just like, let's guess that f(x) = e^rx lol

[@pbj4184]

Well that is how you get the characteristic polynomial right? No need to feel cheaty if you'll learn the derivations eventually :)

[@jkid1134]

Nice and inspires thought

[@a.osethkin55]

Super!

[@thayanithirk1784]

@8:32 IST 611 likes 0 dislikes👌

[@jackychanmaths]

3:35 lower bound?

[@Tehom1]

Your new outdoor setup is interesting but unfortunately makes it harder to read the blackboard.

[@Mathcambo]

Hello everyone. I have a lot of good math exercises for you all.

[@failsmichael2542]

Is this related to Grönwall’s inequality?

[@santiagoferreyra5573]

really nice

[@sil1235]

Just a detail but you said couple of times the derivative is positive and the function is increasing, but in fact in this case the derivative is "just" non-negative and the function is non-decreasing.

[@schweinmachtbree1013]

I think a lot of people sometimes say "increasing" instead of "non-decreasing" just because it's shorter, but you are right about positive vs. non-negative

[@sil1235]

@@schweinmachtbree1013 you are right, I forgot it is sometimes distinguished as increasing vs strictly increasing.

[@mcwulf25]

I can read the chalk better inside

[@abdlazizlairgi9690]

I like the view

[@Veggie13]

Something's different...

[@binaryblade2]

Why can't we set the equation =k for all k >0?

[@nikoladjuric9904]

Because it can be any nonnegative function like k*e^(qx), k(x²+x+1),k(e^x+x²+x-1), for positive k, x>=0, etc.

[@charlesglidden557]

but f' is 2x when F = x**2 not 2x**2

[@BoringExtrovert]

Hello Michael, I think a whiteboard is better for this setting. I am having slight problems telling colors apart.

[@LOL-gn7kv]

Bob Ross does math?

[@jermeekable]

Can we get some more pure pure theoretical focus here and there? Like maybe a run through of Godel's Theorems? Most of these vids appear to be niche solutions to equations, which they are great! Im just saying id like to see some like proofs that go out a little further than the corresponding class itd be in. Anyways still great vids!

[@elie.makdissi]

good point am with you👍

[@jermeekable]

@s s Ill take some topology.. whether algebraic, differential, or vanilla :)

[@elie.makdissi]

@s s yeah if you want, but rather like theorems and equations and identities, you know, like the beautiful ones and rare ones, in topology for example there is the beautiful identity of Euler like V-E+F=2, you know, or to give some interesting facts that can apply to more than one problem, but not like this video, because the solution is niche and specific to this problem not to many more, it look like this imo. But don't get me wrong i like this video and it was well executed and fun, but I'd rather gain knowledge or some facts that my friends or my teachers don't know for example.

[@guruone]

And that's a good place to ... have a BBQ :PPPPP............ Guru#1

[@bopaliyaharshal2399]

Plz see my full comment

[@carterwoodson8818]

In the woods today??

[@michaelslack8900]

This was a really cool problem. Never really considered how much you can achieve when you're not working with an equals sign in the middle

[@GKinWor]

wtff