Solving Homogeneous First Order Differential Equations (Differential Equations 21)
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- เผยแพร่เมื่อ 2 ต.ค. 2024
- / professorleonard
Exercises in Solving Homogeneous First Order Differential Equations with Separation of Variables. A review of various integration techniques is included.
[58:36] "I hope I'm not going too fast for you. If I am... slow down the video" 😂
As I am not able to donate on patreon I watch the full ads I hope it contributes.Thanx for awesome videos
After seeing this comment i disabled my adblocker
@@seaque. i refreshed the page
@@anim8dideas849 i open the ads up
@@sashamuller9743 i buy the products on the ad page instead of donating on patreon in the hopes they will continue advertising on his vids
@@gfdfggghfg2965 I buy stock in the companies themselves in the hope that they'll continue advertising on his vids
Four years ago Prof.Leonard's lectures saved me, but now unfortunately mathematics is not my major subject, but I still listen to his lectures when I get time. I still suggest a lot of people, your lectures for mathematics.
What did you change your major to?
Wow you are 10xE1000000 better than my teacher. She made a 10 minute “lecture” and said good luck 🤦♂️
Big relate
Again, I'm a big fan of those exercice videos. Trying to do them then seeing you doing them really helps
Example problems are definitely the holy grail to learning something properly.
If I ever become a math teacher (and I intend to become that one day) then I will focus a lot on example problems and explaining each step in detail.
How the hell does professor Leonard find the time to be a Uni Professor, be big and muscular and have a wife and kid... simply out of this world...?
Holy moly thank you so much for explaining things step by step unlike every other tutorial out there
dy/dx + x/y +2=0 solve this problem
58:36 I hope I'm not going too fast for you.
in same time, I am watching 2X speed and fast forward all algebra steps
at 30:50 is that the final solution cant it be simplified further???!
These are great, but why don't they keep going. I want Laplace and PDE's!
Thank you for giving me the practice I need to get this down, Professor Leonard!
I don't know Why I spend my all time in searching for differentials equation..Just got your videos...Ahhh the concept I have taken from you in Calculus 1 and 2,just wanted to say thanks Alot..Stay happy and blessed Sir❤
Professor gainz
taking this course right now, Leonard always helps! Unfortunately my class is fast paced and we already covered chapters beyond the videos, currently on eigen values and vectors and fundamental solution sets. Looking forward to the next videos!
wow what are you doing now
This video is amazing, helped me to get the homogenious equations down, i wasnt able to do this on one take though, greetings from colombia, thanks for your amazing content.
Is anyone come here for his biceps?
*Triceps haha
@@agustinmorlino8544
The triceps muscles are actually extremely important - they take up 2/3 of the arms, so anyone who wants big arms should focus a lot on training those muscles - a few good exercises for those muscles are Bench Press, Overhead Press and Barbell Row.
@@Peter_1986 haha, yeah agreed. That's why I said triceps; I was going to pretty much comment exactly what you said but I didnt want to be too technical lol.
These Lectures are very helpful, textbooks are a lot of work!!
Sir, do you have lectures for Metric Spaces ? Thank You
Professor Leonard can you make for more videos in different topics or sub. in DE? I think you're the one who makes me more stay motivated thanks a lot :)
You have helped me a lot in calculus. I am soo grateful and thankful for your lectures
Lessons learnt about integration techniques (some of the nitty gritty details, in particular about U sub)
1) Go through a checklist of methods to integrate before jumping into the tough ones (eg trig sub)
[Basic*>separate into fractions which are basic**>U sub>trig sub]***
*can include other techniques other than U-sub like integration by parts, and 'reverse chain rule' (which is really just another special case of U-sub)
**I only know two ways: first way, when it's one big fraction where numerator has multiple terms while denominator has only one. Second way, by partial fractions.
***if you include the techniques learnt so far, then the list continues from trig sub: [separable>linear ODE>homogenous]
2) All about U sub techniques
i)Hate negative signs! Factor them out, and hope you'll see a repeated expression (eg -x^2 -1= -1*(x^2 +1))
ii)Sum of fractions inside a radical? COMBINE those fractions into one (unless...It integrates nicely into the inverse trig or that logarithm one in your table of integrals)
iii)Non-basic integrals with radicals (eg sqroot U) as a factor in the denominator? FORCE that radical to be factorised out (eg sqroot U - U= sqroot U * (1-sqroot U))
iv)Sometimes you need to do U-sub twice! (Or maybe more)
v) When inspired, feel free to change the U substitution and try integrating again.
Some timestamps
56:56 add fractions that are stuck inside a radical. Then distribute the radical/fractional exponent. (got stuck at that step for some time looking at my integrals table thinking the integral would be arcsin, arccos or arctan)
59:01 be systematic and go through a checklist of the simpler techniques of integration before jumping into difficult ones Saves time. (Basic>separate into fractions which are basic>U sub>trig sub)
1:14:28 U-sub. To figure out what U should be, rewrite your integrand till u get a repeated expression (eg -x^2 -1= -1*(x^2 +1))
1:16:05 Non basic integral with radicals in the denominator. FORCE the radical to be factorised out.
1:16:11 What other way??
1:16:53 Sometimes you need to do U sub twice. (W=sq root U)
1:17:53 I see it but I wouldn't if not for this video
1:18:28 When inspired, you can always change your substitution! Anything that makes the integral simpler (W=1- sqroot U)
1:19:58 a quick summary.
Where does this man teach? I want to go to his school and be in his classes. He has saved my ass countless times.
Professor Leonard, thank you sooooooooo much about you did in statistics lectures. The videos are so great and helpful. Do you have for chapter 9.10.11.12 ?
These are mostly algebra and just a little calculus. I admit I usually didn't make it through all the algebra without making an error!
I think the first problem you do has a solution (30:51) of y = (Ce^(x/y))/x. I went through it twice and got the same answer. If you use the substitution y=vx then y'=v'x+v. You can substitute both into your given equation, without having to manipulate your equation to get fractions in form of y/x. Please check that as I'm not 100% sure.
thanks so much for being one of the best at what you do , you're are unique and I hope you only get better , amen .
Hi guys you may test the lecture which is given below the link
th-cam.com/video/qUmLfJP3BaY/w-d-xo.html
I'm a straightgeneous.
Question: if we divide everything by X, why don't we do the same for the left side? As in the result would be (1/x)(dy/dx)=...
1:33:03 You can't _just_ separate the fractions, but you _can_ just separate them anyway 😏, get two fractions, simplify the denominators into v(1+v^2), and use the u-sub u = (1+v^2). This makes the first fraction immediately solvable (but we leave it for later because it can be handled at the same time as another very soon), and with the second, using the concurrent u-sub v^2 = u-1, we get our denominator into a form where we can use linear partial fractions. Then everything cleans up nicely.
(Just an alternative for those who didn't recall how to handle the quadratics in PFD or didn't see that as an option first.)
that wouldnt work because when we do the derivative of (v^2 + 1) we get du = 2vdv and du is dv= du/2v and we cant simplify the v that is in dhe denominators
i hope you see this and understand what i mean
@@OlsiDisha But it does! To see this, consider that after moving the constant 1/2 from dv = du/2v outside the integral, you have (vu) ⋅ v, that is, v²u, in the denominator of both fractions. By our u-substitution, v² = u-1, so our integrand is in terms of only u. You then proceed as above.
@@osianoisekenegbe9401 yeah now i got it thanks
That's what I call no-holds-barred bare knuckle calculus.
getting railed like this only works when you have someone to explain the problem and allow you to see your every mistake, clearly.
Sir, at 53:22 , can i do sqrt(4x^2+y^2)/y = sqrt(x^2(4+y^2/x^2))/y=|x|/y * sqrt(4+y^2/x^2)=plus or minus 1/v * sqrt(4+v^2)
I think I would understand this more if he was shirtless. Just saying.
Clark Kent switch job to become a math professor ??? neat.
Please help me with this e^xcosy-2x+(e^xsiny+1)dy/dx=0
A bunch of students are waiting you to upload laplace, second order come on professor you can do it
We'll get there, but I have some other videos I need to do first.
hocam burdan çalışıyorum derslere de, acaba ilerleyen videolarda laplace ı ekledi mi ? bilginiz var mı
Awesome Job! The detail to attention is extraordinary!
What is a good differential equation text book?
For y(dy/dx) + x = sqrt(x^2 + y^2) Couldn't you do this???
dy/dx + x/y = sqrt(x^2 + y^2)/y [divide by y]
dy/dx = sqrt([x^2 + y^2]/y^2) - x/y [subtract x/y to the right side and expand sqrt to the denominator]
dy/dx = sqrt(x^2/y^2) + sqrt(y^2/y^2) - x/y [separate fraction]
dy/dx = x/y + 1 - x/y [y's cancel to 1 and take the other sqrt]
dy/dx = 1 [simplify]
dy = dx [separate terms]
then just integrate from there???
I know of another teacher who describes a technique called homogenous but it looks very different than yours. What would you call the following technique and are they connected in any way?
(D^2 +4D + 5)^2 * X=0
(D^2 +4D + 5) * (D^2 +4D + 5) * X=0
(r^2 +4r + 5) (r^2 +4r + 5) =0
factor using quad eq
(-2+i) (-2-i) multiplicity of 2
x(t)=(Csub1 e^-2t) cost + (Csub2 e^-2t) sin t + (Csub3 t e^-2t)cost + (Csub4 t e^-2t) sint
Thanks for any replies!
@44.43...Was always taught this as differential of denominator over denominator...now I understand why...comes from a u sub. Thanks Professor.
Sir you look like actor Henry cavill 🥰😘😍😊 like superman film hero
when are the higher order videos coming out?
27:45 at this moment change -1/v to -x/y and change -2lnx to lnx^-2 then add x/y both sides and elevate to e, you got y= x^-1 times e^x/y times C , there is an y but i think looks nicer
IN DOMAIN SECTION
EXAMPLE 1
WOULD IT BE TRUE TO SAY THAT (y/x is not equal -1)?????
Could you solve in rectangular form with y as sin theta etc. You get theta minus ln sin theta plus c.
much thanks! the questions you ask help understand how to think about it
Maths teacher i dreamt teached me back then
Would you please make videos about linear Algebra. that would save many many people.
did he say "porque" im dead hahahaha
Pls can u explain how ∫e^vdv turned to -e^-v
Thanks a lot for these examples.
Alright very helpful. Please release gym split next.
Do these equations right and you are ready to serve the First Order 💪💪
please do REAL ANALYSIS IM GONNA FAIL SO HAARD
lmao jokes on u i passed with D- ;)
Kaan goodjob!!
@@classic4624 ty buddy , it was a rough time
I am very much interested in your lectures and it also helped me learn American accent. I don't even find words to thank for!!!!
Trying to concentrate on teaching ...still watching a burning dude
At 50:10 why is the explicit solution not recommended
A million Thank Yous !!!!
الله يهديك ويصلح لك شأنك ويزوجك يخي، لس ساعتين ع السؤال 😂
متزوج الله يوفقه ويهديه
@@ie6730 عجل الله يرزقه بالبنين والبنات، الحمد لله خلصت الميدتيرم اليوم 😂
in the first examle, shouldnt you devide the left side also by x
Thanks professor
Thank you so much for posting these videos. They have helped me considerably. I just wanted to note one thing for you, although it wasn't too bad in this video, some videos you lose me with constant recaps. Sometimes it helps to have a linear train of thought straight to the point. And doing recaps really quickly is very hard to follow when you still don't really get the process. That being said, having them after explaining a concept fully is really helpful. Just in the middle in becomes quite confusing and disrupts your train of understanding. Thought I would share how I feel as a student since you cannot gauge any reactions from us. Cheers professor, your doing gods work :)
can't you just factor out the negative in 1-v? so it's -(1+v) and cancel with the denominator
no because if you were to factor out the negative it would become -(-1+v). sorry if you dont need this anymore but i just thought id comment.
ugh i did the first problem integral after having everything in v by completing the square and got something else man im rusty on integrals
keep it up, sir
1:04:04
is there a video on the power series method for these?thx
THANKS!
these DE's arent that bad... recognizing you have a homogeneous DE is tho... :l
yes, it certainly can be!
sighhh.... many years late... i wish i had these awhile ago. Thanks for posting your content online.
Do you do non-homogenous ODE's? Would love the link to that video...
BTW the videos you make I use for my grad classes because you explain so well!
18:36
5:00
How at 1:24:06 can the abs-value be dropped when in the previous steps non positive values are undefined in ln|(1-sgr(v^2+1)| ? infact the enclossed valuse never postive, but always negative or zero.
He only 'dropped' the absolute value signs, after he anti-logged (is that even a word hahaha)!
Sometimes BEFORE the natural logarithm is 'removed' (meaning you take the inverse operation on both sides of the equation, that is, raise e to the power of the LHS and the RHS. Which is what I meant by anti-logged)...
...the absolute value signs can just be removed/replaced with ordinary brackets, if the expression inside the absolute value signs is strictly more than zero. (And yeah I agree with you that the expression 1-sqroot(v^2+1) is not always positive in this case (in fact it's less than or equal to zero for all x), so he's not allowed to just drop the absolute value signs)
But in this case, the prof didn't do that! He was just going through the normal process of removing the log. Then opening up the absolute value signs (the plus minus sign on the RHS).
PS I edited this comment a few times hopefully I got it right now hahaha
You are great Prof! Thank you!
31:01
thank you so much
Sir your way of teaching is amazing.
When is your Birthday Sir??
great job
Wow !
Thank you 🇮🇶
hot
ما خوش تدريس
Sitting here, getting ready to practice these old techniques. Heyo, I love you my man. Other than math ive been lifting for 2 years now, and my biceps are starting to look like yours!
Jesus these problems are absolute nightmares.
Thank you great sir!
This is kind of help i needed.
also in 51 minute, i just took the square of 4(x^2)+(y^2). Which left me with (2x+y) and then i multiplied with x in the question and then i solved it.
_guess i did wrong..._
You're my hero.
Thank you sir.....😉
Thank you
In your opinion, which text book best teaches differential equations?
Instead of u=1+v^2 couldn't we substitute u^2=1+v^2?
iN 1:02:00 WE CAN TAKE THE SQYARE ROOT FOR BOTH SIDES AND GET RID OF THE SQUARE RIGHT ?
nope! you can take the sqyare root, but not get rid of it
@@xnorgate5894 we take square root for both sides and we take out the power fro both sides and then we left with square root of 4
those muscles sir makes us understand less
Leonard .sheldon😁
1:16:11 Anyone knows another way to solve that integral?
you can not substitue but you will need partial fractions
So you split it into A/1-sqrootU + B/sqrootU ? Never did partial fractions with radicals in the denominator before though
Are there any limitations to partial fractions??
I got A=-1,B=-1
@@khbye2411 Yeah you can do partial fractions with radicals. It is essentially the same as substituting the radical with another variable. I did not do it my self but you can check if A and B are correct by adding the fractions and see if you get the one you started with.
@@kostasmerenidis1307 That's what I did because I didn't think about forcing it to factor, it ended up fine.