Method of Undetermined Coefficients - Part 2

[colonelquack]

And suddenly it makes sense.

[alexloldun]

I haven't been to class for the past 3 weeks, and i've caught up in 2 days with the help of your videos, practice problems and hard work. Thanks patrick!

[mathewrokz]

love it thanks for the help...

[alfscott11]

what if y" or y' have an x in front.

like xy" +bla bla bla, or 3y" + xy' +bla bla bla

[omogaju]

why would any one dislike the video?? either you watch and learn from it or you don't watch. Perhaps YouTube should start revealing the screen name of morons that click dislike to a video such as this.

[patrickJMT]

@omogaju ha ; )

[UWGuy2013]

taking my ODE final tomorrow. and this is the best help that i have recieved all semester. the best thing is that you actually speak english

[1fyefye]

ugh !!!! thank u sooooooooooo very much.. this was a great help....u eased my stress.. God bless!!!

[megabenman]

PatrickJMT: Y U NO DO VARIATION OF PARAMETERS?? :(

[taeahmed]

Thank you PatricJMT for your videos i really learned a lot, but i have a question.

What if there isn’t a form that i can use such as this problem:

y” + y’ = constant e^constant x + constant x

What i did in this case i differentiated the constant x and took it to the other side of the equation is this right???

many thanks again

[Figindulgent511]

I went through two textbooks & they couldn't explain things the way you just did =.=

[Rush2112CM]

Thank you so much your videos are helping me understand everything we learned in calculus! I was completely lost before i watched your videos :p

[mynorperez]

How do we know when to multiply by X before taking the derivative of the particular solution? I know in some cases it will cancel out because it needs the extra X. Is there a formula that will explain these certain cases?

[mrpwnr24]

Thanks for the help and clear step by step instructions

[Phacix]

thanx a lot, it was very clear to me

[leocheyz90]

dear patrick,please make non homo equation-Variation of parameter tutorial for me. . i need it so bad. .

[jesusgalarza1]

What is the best way to guess what the particular equation could be?

[GamerJimbo]

Could you show an example of a nonhomogeneous linear equation problem that requires you to multiply an x to the y_p so that it would not look similar to the y_c?

[mynorperez]

@GamerJimbo That's what I need help with too! I don't understand when to multiply an x or x^2 to the particular solution so it won't cancel out.

[honeypot11]

i have to say, me and my friends are always so happy whenever we find that you've made a video about what we're learning. Its like:

"Oh man that was so confusing"

"Its ok, patrickjmt has a video on it!"

"YAY"

lol

[patrickJMT]

@honeypot11 ha, sweet! : )

[latecomings]

i pay 100$ for cramster and 32,000$ a year for tuition and thanks to this man with his free youtube videos I don't waste $32,000 a year.. thanks man ! I probly would of failed every math course without you and woulda wasteed 32k a year and drop out of college and become a bum but instead im almost an engineer now

[patrickJMT]

@latecomings happy i could help you out : )

[bmpaquette]

you should be getting a portion of my tuition

[patrickJMT]

@bmpaquette donation link on my website : ) feel free to send all of it if you want!

[hey0its0nick]

Patrick, you are the Bruce Lee of math. Words cannot express my gratitude for these videos!

Keep up the good work!

[patrickJMT]

@hey0its0nick more like chuck norris since i am as white as white can be! : )

[SuperPowderkeg]

Dear Patrick, how would you solve a problem like so: y''- 7y' +12y = 170sinx when you are asked for the particular solution?

What can be done about the 170sinx?

[Darathexplicit]

you are possibly the most thanked,loved and complimented person on youtube

[adamwitt10]

Mate you're an absolute champion. Nice easy to follow instructions, and a voice that isnt like every other typical boring old fart mathematician that lives at my uni.

Keep up the good work!

[tattleboad]

@voidwareprohibited i'm not i just happened upon this video, i'm a history major so all i need to know is college algebra

[tattleboad]

god i hate math

[alexwhb122]

This was an awesome explanation. thank you!!! this helped so much!

[sirjim2009]

You are brilliant

[alaaddin477]

ooh my brother thanks a lot. you teach us better than our professors even our teachers who speak my native language. thank you god save you.

[KinKyBuzzingFM]

your such a good teacher u should get a world prize for the best teacher in the world!!

[defyzai]

How to find the tricky part ? is there a list of formula for reference ?

[TKollaKid]

I was just studying organic chemistry when I decided to take a break. Now Im on youtube watching math videos. lol Im a nerd :(

[rimzo2006]

i hate equasints, cpos it in the eye of hte behoder,for eg i have no idea why your a1 = one half, how does 1 = 0.5, the rest makes no sence, hope this get u layed :P lol

[Rizmaestro]

Hey great vid as always patrick, but I was wondering if you could give us a Csinx + Dcosx example when solving the particular integral?? That would really make my day!!! Keep up the good work!! From Scotland :)

[luumanh123]

hi, how can you imagine the yp(x) when the f(x) is some thing like cos(x), thanks

[BenBown]

What would I do if my G(x)=e^-x ? how can i find a particular solution for that?

Help please? :(

[eminemineminem]

Hey thanks for the video, its really good. Only thing is that i dont understand the fuess part, how do you guess a Yp just by looking at Yc, i mean in detail.

[extremecoola]

I love ur maths vid! help me loads. thanks,

just wanna know do u have videos on variation of parameters method to slove 2nd ODE?

[SuperPowderkeg]

Gr8 vid, 5mins with Pat = 1 year with lectrurer!

[tucense]

I hate the unstandardized vocabulary of math.. There are different names for everything with the way I learned. UC sets. things like that.

[MikeyRGuitar]

jeese, I can't understand for the life of me why this is the only guy who can explain this logically. I searched for a day trying to figure this out, thanks man.

[BazteK7]

Yea I'm a Mechanical Engineering major watching since year 1. But for my differential equations and linear algebra class, I couldn't find any of your videos at the beginning and just remembered these exist. A little late cause good chance I'll have to retake it but if no biggie I'll live.

[xxSimplySoniaxx]

YOU have a gift!!! I hated all my math teachers till I came across your videos. I only wish my school would stop using books and would refer us to your channel instead!!!

[Haroson]

Would it be cool if you post higher order linear differential equations where the particular solution is a 3rd order polynomial? Please? Thanks :)

[patrickJMT]

@ProMysteryGuy happy to help. good luck in the studies!

[osamamaqbool]

why dont we just plug ax^2??

[Minandreas213]

Honestly, why do I have to pay hundreds of dollars for each of my math courses? With guys like you around, I am basically paying all of that money just for them to stress me out with a test 4 times a semester and give me a grade. Out of all my math classes I have yet to find a teacher comparable to you. They should just send 90% of my tuition your way. You've gotten me through calc 1, all the way up to DE now. Thanks a ton man.

[4thKyuubi]

Do you have other examples of this? Examples like -3te^(-t) other than x^2

[mikester9er]

can this method also work for a first order ODE?

[Demagogue88]

Oh fond memories... This reminds me why I love math and makes me happy to know LaPlace Transforms as well as method of Annihilators. Still, this is rather awesome for the casual review every now and then.

[Demagogue88]

Oh fond memories... This reminds me why I love math and makes me happy to know LaPlace Transforms as well as method of Annihilators. Still, this is rather awesome for the casual review every now and then.

[jessemaurais]

What about C1 and C2? I've answered similar questions with values for C1 and C2 made to fit the homogeneous equations, but this is apparently incorrect. If you give Wolfram this

y'' + 10y' + 24y = −120t^2 + 20t − 8, y(0)=4, y'(0)=-29

and then this

y'' + 10y' + 24y = 0, y(0)=4, y'(0)=-29

You get different values for the coefficients of the general solution. My textbook does not cover this, and I haven't yet found a video which does.

[DirrrtyDutchMTRFCKR]

@jessemaurais Idd, what about C1 and C2? How do you find those? Don't you need to know those initial values for that or something. Like y(0)=0 or y'(0)= 1

[Eireboi1]

Very good

[shaoyuan]

hey how did you get the x^2 to become 1x^2 + 0x + 0?

[abdullah57habib]

@shaoyuan He's basically writing x^2 in a different form. So 1x^2 + 0x + 0 = x^2. He's doing this so he can equate the coefficients

[DirrrtyDutchMTRFCKR]

@shaoyuan He wants ax²+bx+c to become 1*x². So you have to take a = 1, b = 0 and c = 0.

[RaySolava]

Dude your videos have saved my ass countless times before midterms and finals......THANK YOU SO MUCH!!!!!

[kebabsallad]

Today I went up, made myself a cup of coffee, took it slow and sat down here to watch your vids on differential equations

Yesterday I went up, stressed to get dressed, ran out in -17 degrees celsius pre-winter to catch the crowded train, on which I spent 35min to later arrive in the city. There I took the tram on which I spent 10 more min, then I arrived at the university, sat down at the lecture 4 2h, learned this:

PatrickJMT is one hell of a teacher, I choose him over lectures!

[no1everreallydiez]

If he wrote a textbook, I think he'd be as rich as Stewart.

[Haroson]

Do you have videos of solving equations involving trig or polynomials because those are the nastiest :/

[jackbenimble7]

These damn problems take so long and are extremely tedious...especially those with more complicated f(t)'s on the right side...

[jemmandthe]

Ur vids rock!! I was wondering could you do an example where the equation is just equal to a number like y'' - 6y' + 8y = -24?

[emanueluke]

Someday textbooks will be a thing of the past, and there will just be a playlist of your videos.

[patrickJMT]

@emanueluke ha, that would be the day! : )

[StrawberrySong]

All these mathematical processes are so simple, but I can only see that they're simple once I watch your tutorials. Thank you!!!!!!!

[bgates1128]

Thanks. I needed this.

[MachViper09]

Oh, and I forgot to mention this. In my classes, you're like a celebrity. Whenever we figure out something difficult and share how to do it, most of the time we end up saying "Yeah, patrickJMT did this on youtube, made it totally easy!"

[patrickJMT]

@MachViper09 ha, very funny : )

[MachViper09]

You contribute more to my knowledge of Calculus and Differential Equations than my instructors. Watching you do this is like magic. Textbooks be damned, I want patrickJMT!

[sandysanchez4]

i don't understand how you decide what to choose for Yp. For these examples it seems pretty easy but for others i don't know how to choose.

[sixstillys]

That was cool. Do you have any solutions which have multiple trig on the RHS such as y"+4y=sinx+cosx?

[juanpablodonche]

amazing

just great

i hope you make class in my college

that would be so nice

[juanpablodonche]

amazing

just great



[ppingpoong]

i love how your videos are 3 minutes, because thats usually my attention span i have in lectures

[patrickJMT]

@ppingpoong mine too!

[nixrate]

Thank you so much. You covered the exact question I was having problems with. I had it right up until solving for B which needed to be rearranged. Thank you for the beautiful explanation. :D

[PseudoScienceb20]

Just an observation: Say why you erased the "C" term at 2:52.

[JeneralJones]

@PseudoScienceb20 probably just an accident, I think the C is supposed to be there

[bigluser]

This man is a saint!

[Finda82]

thank you so much for this lecture. I really appreciate it. I was unable to understand the process in class but after watching this video; it makes so much sense. Is there a video for solving the same kind of problem but given an initial condition?

[Finda82]

thank you so much for this lecture. I really appreciate it. I was unable to understand the process in class but after watching this video; it makes so much sense. Is there a video for solving the same kind of problem but given an initial condition?

[dhemi1]

This video helps a lot thanks Patrick!

In my Diff EQ class the professor used what was called an "Undetermined Coefficient Function" to get the polynomial to solve for the As Bs and Cs

i.e. x^3---> {x^3,x^2,x,1}

sin(2x)-----> {sin(2*x), cos(2*x)}

e^(7*x)-----> {e^(7*x)}

its taking all the possible derivatives of the function whilst ignoring the coefficients.

[dhemi1]

This video helps a lot thanks Patrick!

In my Diff EQ class the professor used what was called an "Undetermined Coefficient Function" to get to solving the As Bs and Cs

i.e. x^3---> {x^3,x^2,x,1}

sin(2x)-----> {sin(2*x), cos(2*x)}

e^(7*x)-----> {e^(7*x)}

its taking all the possible derivatives of the function whilst ignoring the coefficients.

[BearzBearzz]

I skipped all my lectures and juz study by streaming this 9 mins video... U rulez Patrick!! Summarizing a 1 hour lecture to 9 mins and in an easier to understand way!!! THX A BUNCH XD

[jieun7543]

Does anybody know what to guess for an equation like

y'' + 16 y = 8 sec(4x) ?

I know for the equations involving sin x or cos x

(like y'' + 16 y = 8 sin (4x) )

I would guess y(p) = A sin (4x) + B cos (4x)

would i use sin x cos x for sec x as well? since sec x is like 1/cos x?

[TheBucksB]

very nice, thanks again

[guicapone2]

why is the recorded sound low.. can u plz speak to the microphone closer next time thanks..going HD is not bad.. but u left the sound behind..

[MadCow582]

Dont include any terms that are already in yp." I get rule 1, but I'm having a hard time figuring out rule 2: Hopefully you'll be able to understand it better than i can, and hopefully explain it. Thanks

[liquidstl]

@MadCow582 Im not entirely sure what to do when there is duplication between Yc and Yp either.. I know you need to multiply Yp by some power of X so that it doesnt appear in the complamentary (auxillary) solution, but my book also says that Yp's derivatives cant appear in Yc either... and im haveing trouble applying it

[MadCow582]

@liquidstl Yc I'm assuming is the general solution; It should contain Yp (particular soln) and the Yh(homogeneous soln). It's just that when you apply the derivative, whenever you get an answer that is the same as Yh soln (by arbitrarily setting your Constants to obtain), you skip over it.

[MadCow582]

Rule 2: Suppose that a term in F(x) is of the form x^n f(x) (n a nonnegative integer). Suppose further that f(x) can be obtained from yh(x) by specifying values for the arbitrary constants. If this term in yh results from a root of the auxiliary equation of multiplicity k, then corresponding to x^n f(x), assume as part of yp the term Ax^k (x^n f(x)) = Ax^(n+k) f(x), plus a constant multiplied by an linearly independent function arising from it by differentiation.

[MadCow582]

My Engineering Calc has these 2 rules to help lessen the load for find the particular solution. Rule 1: If a term of F(x) consists of a power (x^n), an exponential (e^(px)), a sine (sin px), a cosine (cos px), or any product thereof, assume as part of yp a constant multiplied by that term plus a constant multiplied by an linearly independent function arising from it by differentiation.

[ranniskurast]

THANK YOU SO MUCH!!! Passed my mid term because of this :) :) :)

[Ronaldoh1]

you're awesome!!!!!!!!!!!!!!!!!!!!!!!­!

[achironis10]

could you do a variation by parameter's example using determinants?

[lahire149]

thanks for getting me through Georgia Tech

[Tseles2000]

Why hadn't I seen these videos so far??

Maybe this time I 'll pass my exams..!

Great work by the way!

[turtleman1234]

Thanks a bunch, I am terrible at calculus but this certainly helps.

[4CMil4nf4n]

You do an amazing job. This helps me so much. I agree with tuberALEX though, I would love to see some more difficult examples. Because while I can understand the simple problems, the longer ones when you don't set it equal to a polynomial are more difficult. THANKS AGAIN YOU ROCK!

[RICbass78]

hahaha 8:28 "Kind of a long process, but um...... well, I dunno, it's a pretty tedious process"

lol yes, yes it is.

[roloaguilera]

i would like you to include undetermined repeated coeffcients

[eztropheus]

Man you rock.

Eztropheus from Mac

[f1f2f3f4f4]

your are da man! great stuff

[TheBlackSunTVProduct]

Thanks so much!

[tuberALEX]

pat, you should consider showing more challenging examples

[JeneralJosh]

@tuberALEX I second this.

[li10ne]

thanks again!!! u make it look sooooo easy...you are the BEST!!!

[fyghetr]

your are awesome teacher, i would like to haw same rector as you ^^ ty

[antlion3]

I was thinking that you should also post a 3rd part on special cases. ie. reapeating and imaginary roots fro the general solution.

[dd02caca]

hi..thanx for the video..its realli helpful

but i got stuck on one could u help me out?

its

y''-7y'+12=xe^(2x)

[tanvec]

I assume you mean y''-7y'+12y = xe^(2x)...

try a particular solution of the form Ax*e^(2x)+B*e^(2x) and follow the process outlined in this great video

[widewombat]

Lol I was getting a headache trying to understand this method just using a textbook. Now I'm getting somewhere. The way you go through things is really good, thanks for the vids!

[patrickJMT]

happy to help : )

[marlesus]

Thank you very much, patrickJMT! Your videos are VERY useful! Best wishes from Romania!

[guicapone2]

@widewombat

lucky.. i still don't get it either way :(

[xpar4n01a]

thank you very much

[shineforoursaviour]

by the way, have you got any videos explaining how to solve a question involving Riemann sums, where you have to change the product into a sum?

[shineforoursaviour]

omgg you should be replacing my math prof at the uni!!!! :))) thank you very much for your videos! ive been trying to understand this, and this video DID help alot! pls post more examples under this same topic if you can!..you're gonna be the reason for most of our 'succes' at the exams! =D

[MPV839]

hi patrick...what equation would I use for y [sub p] at the beginning if the original problem is y'' + 6y' + 9y = 1 + x?

[kasdkijhnijf]

what if you said yp(x)=Ax^2 so when you solve A yp(x)=x^4/(2x^2+6x+2). Is there something wrong with this answer because i dont understand why you choose Ax^2+Bx+C when you could just choose Ax^2.

[patrickJMT]

well, the point is that my second degree polynomial is more general!

[kasdkijhnijf]

What do you mean by that? That its in a simplier form and it looks nicer. Would my answer be wrong?

[patrickJMT]

i am not sure, give it a try and see if your works and let me know.

[christopherchubb]

It wouldn't work because its derivatives wouldn't satisfy the equation for which he was trying to make it. If it worked without only an x^2 component then he should have found that B and C were both 0, but they were not, so presumably the coefficients he found were the only applicable ones, in some situations you may be right, but in general I'd say it isn't safe to assume

[kasdkijhnijf]

Thanks

[henricx1]

wow

this is very usefull

[fldoughboy72]

Interesting stuff. I never did this before. I never got to take diff-eq in college. I stopped at Calc 3.

[patrickJMT]

it feels like calculus 4.

it is really really useful and really really cool stuff in my opinion. i wish i understood more of it

[eedahl]

I like that you're putting up more advanced stuff as well. I can't really even tackle the easier stuff, but I like to glance at what I might be doing some 2-3 years from now, and it makes it easier to read books way over my head when I know what the heck they're talking about :-P

[RapidFireAlgorithm]

what is the simplest procedure to calculate the GS when the coefficients are functions of x?

eg:

x^2(y'') + 4x(y') +2(y) = 0

If you have already shown this, would you point me to that video please.

thanks in advance.

[daryl31415]

i think you need to use the Cauchy-Euler equation, if that helps :-)