Linear versus Nonlinear Differential Equations

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Linear versus Nonlinear Differential Equations

[@Scarabola]

Why are books and college teachers so bad at explaining these concepts? I swear.

[@tauceti8341]

To be fair I think its the same reason we can't explain so basic algebra concepts to students. What we might be able to see as understandable is beyond grasp. Also there's like 5 different learning styles or something, visual, kinaesthetic, and some other stuff. That being said. There's no one universal model for learning which is why open resource learning will be the key to the future of learning and education, and should have been from the start of the internet. Now we get 250$ required textbooxs that explain bollox while the real heroes are going unpaid but because they love maths and love teaching like Math Sorc. Just throwing out an idea, but I humbly agree Bulldog, the model makes me mad, so I rather get mad at not being able to do maths! :-P I swear though, that earlier texts like Dover, and papa, are great because the authors actually had the enthusiast in mind, not the 'student' which I think is a big difference. Since my local bookstore is gone I no longer have access to those texts for pennies on cent's. I'm incredibly grateful to have a mathematic resource like Math Sorc, when a majority of maths majors are there to make big checks in finance/insurance/buisness. My dude likes pizza though, and that's all that seemingly matters. And hopefully some cool surfboards XD

[@TheMathSorcerer]

👍

[@kevinmurphy4946]

The teachers are good for my part honestly

[@josephdillon5540]

Tell me why my professor literally said that they are linear if they are “pretty” and just moved on 😂😂😂

[@Fleato]

Cause it’s very wordy and if you don’t understand any single point thoroughly you’ll be lost. Like I am.

[@hydex2885]

Man studying for this DE test without a textbook is a PAIN. You're a life saver. Great video

[@TheMathSorcerer]

Welcome😃

[@doosen.]

My MAT 275 professor did not really explain the difference between linear and nonlinear diff-eqns so I'm really glad I came across this video. Literally helped me understand this topic in 7 minutes compared to the 30 minutes or so he spent going over his pre-written notes during lecture on Tuesday.

[@TheMathSorcerer]

👍

[@omnibrain8]

I'm very grateful for watching this video. You have cleared out the confusion in my mind. I was surprised my answers to your examples were all correct. I owe you millions of thanks. Thank you so much.

[@TheMathSorcerer]

Glad it helped😄

[@sofiadale9015]

thank you so so much. Im taking diff. equations and i was so lost in lecture. this was awesome, such an easy explanation and great examples :)

[@TheMathSorcerer]

You are welcome!!

[@user-ii4pq8vb4f]

I just finished my class, and couldn't understand my doctor explanation. This is way better, so thank you SO much ❤️❤️❤️❤️❤️❤️❤️

[@TheMathSorcerer]

You are very welcome❤️

[@Dana__black]

My college teacher basically told us read the book, and we’ll meet when there is a quiz or test. It must be nice being a teacher during COVID. They do the bare minimum.

[@akhilgajjala4922]

Flipped classroom on roids.

[@_lu_ferro]

I get that but it really depends on the teacher, some of mine were like yours, but not all of them. In fact, most of my professors really do try hard to teach, they had to basically learn from scratch how to use technology in education. Plus, most students don't interact, which doesn't really help

[@Dana__black]

@@_lu_ferro well no duh it depends on the teacher 🤦🏽‍♂️ lol

[@Ryfinius]

way better than how my book explained it.

[@TheMathSorcerer]

yay glad it helped!!

[@jorgeequihua5159]

Excellent explanation. Much better than the explanation found on books or other videos. Thank you.

[@valor36az]

You are such a good teacher, I struggled with this after reading it in Zill’s book.

[@TheMathSorcerer]

Yes it's so hard to read DE books!!! I will review some other ones soon. Thank you❤️

[@2creativesisters266]

same

[@jinkim1877]

thank you so much you just saved my life!

[@elmsudraws8407]

I'm in middle school and could never grasp linear and nonlinear before, I always needed my notes. After watching this video I understood it a lot more quickly and easily. You didn't didn't overcomplicate things which I think my teacher did. Thank you, this will help me so much my test

[@angelicabendo4708]

i like your voice and tone. Makes me wanna listen to math lectures.

[@TheMathSorcerer]

Aww thank you!!!!

[@bush_0074]

My professor threw a bunch of hard to understand words and terms that came from a textbook and it's such a pain in the a**. This video helped me understand the lesson surprisingly fast and easy. Thanks a lot you've earned a like and subscribe from me.

[@cecejames353]

Hi! I'm kinda confused on what you mean by pure functions of x or pure functions of y? Would you mind explaining it a little more and providing an example? Other than that I really love your content, it has helped me so much with understanding ODE and other math concepts. Lol as a fellow math major I struggle sometimes to understand and really grasp certain concepts

[@AyushGupta-rq4qu]

Pure functions of x would mean f(x) or f(g(x)) not f(x,y) or f(x,y^2).

[@cecejames353]

@@AyushGupta-rq4qu Oh okay, that makes sense. Thank you :)

[@clarayi1840]

can you also count constants as pure functions? like 3, k, 7, etc. @@AyushGupta-rq4qu

[@AyushGupta-rq4qu]

@@clarayi1840 well 3=3x^0. So probably yes.

[@lukeyard100]

This is a great way to refamiliarize myself with this content, thank you

[@TheMathSorcerer]

You are welcome😃

[@Charan-yr9te]

5 years later and it is still helpful.....thank you

[@basheera93]

thank you. I needed the refresh for my system dynamics class

[@TheMathSorcerer]

You're very welcome!

[@yehah]

Hello Maths Sorcerer, if the y and all its derivatives all have to be to the first power why is equation 1 called Linear when the first derivative term in the (a) example is raised to the second power? Seen this in many examples and still gets us. Or is it the case where one of the derivative terms has to have a power of one and the rest can be to whatever power? The pure function of x condition is understandable. Thank you

[@sevgipnar5261]

I also couldn’t understand, someone please help why did we say first power but we took 2nd derivatives

[@shreenthasleem2820]

Only the person who understand well could be able to teach well . This was perfect

[@julianaricci3474]

This video helped more than a week of lectures.

[@priyankapulikeshi6003]

literally the best video for identifying linear differential equations, thank you!!!

[@TheMathSorcerer]

Glad it was helpful!

[@ducksonpan8763]

Your explanation is so good. Thanks for making my life easier.

[@TheMathSorcerer]

np!

[@ducksonpan8763]

If its possible then pls make a video on how to determine homogeneous and non homogeneous equation similar to this one.

[@anthonystevens6999]

Thank you for clearly explaining what my professor couldn't in an hour and a half.

[@joelhogeman7607]

Thank you very much for the help! Sorry for pointing this out but I can't help myself, it's correct to write "its derivatives" rather than "it's derivatives".

[@TheMathSorcerer]

Thx❤️

[@itsmeyaboi6721]

Thanks for explaining it so well! Seeing it in words definitely took it from making no sense to being pretty easy, haha. Only seeing the mathematical definition made it tough to visualize.

[@lindsayb265]

best video I found on this topic. Thanks!

[@TheMathSorcerer]

😃

[@dishajindal7853]

Thankyou for explaining this concept. But I have a doubt in the (d)part if we take 3y to the right hand side it would be { √dy/dx =sinx - 3y } and now if we square it we will get dy/dx= sin²x+9y²-6sinx y. And its linear I guess.... Can you please clarify my doubt asap I have my test soon..

[@dantsai5216]

Let's do uh-nother one!!! 4:38

[@TheMathSorcerer]

hahahaha

[@SzechSauce]

Thanks a lot for this, great video!

[@waltermburu2520]

I love the way you teach sir 🙏🙏

[@armiraquibuyen975]

Thank you so much for this. Took months before I've understood. I should've discovered this channel earlier. :((( Subscribed and shared to mg friends! Thank you

[@TheMathSorcerer]

Thank you!!

[@enass.muhammed7469]

Thaaaank U, really!!! I had a PDElecgure yesterday and I was really confused!!! U. Helped a lot

[@mohammedal-khadouri5972]

i really don't see a reason why my instructor didn't explain it like this. thanks!!

[@TheMathSorcerer]

you are welcome!

[@AlexanderWebster_]

It's cuz this dude is a sorcerer.

[@TheMathSorcerer]

Yes lol

[@dsstntech]

my goodness... I read the book so many times over and couldn't figure it out... this became clear in only 7 min

[@TheMathSorcerer]

👍👍👍

[@muggarysagna6994]

Thank you, this video was a lot of help

[@halffreeze2240]

Thanks for the explanations, you are way better than my lect.👍👍

[@TheMathSorcerer]

Happy to help!

[@trektor6673]

Very well explained. I always mixed up between order and linearity

[@metanaito5305]

Thank you this was very helpful!

[@TheMathSorcerer]

You're so welcome!

[@jenieljinabhai4850]

Hi if I have an equation sin(y) = x would that be linear ? and would sin(x)sin(y) = x be linear ?

[@jovanmagtoto1682]

Thank you for this sir, clear explanation.

[@TheMathSorcerer]

Welcome😃

[@TommyCherry]

This helped a lot, thank you!

[@TheMathSorcerer]

You're welcome!

[@mariamkamal5919]

thank you that was really clear

[@nikhaziq1787]

you explained this really well . Thank you very much! You just got a subscriber!

[@TheMathSorcerer]

aww thanks

[@tamurtaj2722]

concise and the best explanation. Blessings

[@mostfunnyvideo6139]

maths socerer make me your apprentice.

[@crehenge2386]

Is this the same course as on Udemy? Any how I'm buying it to support you! You are a great teacher!!

[@TheMathSorcerer]

No it's slightly different 😄 awesome thank you!

[@mohammad_bilal951]

Oh man

[@TheMathSorcerer]

❤️

[@mohammad_bilal951]

@@mohsenhca Shukran akhi

[@donathmm3881]

Good and easy explanation thanks

[@N7TWL]

First of all, thank you for your videos. A couple of questions: (1) What do you mean by "pure funcion?" My Zill text does not have a definition for that phrase. (2) In your example a, you give an equation that has the second derivative of y and so forth. If y must be to the first power, how would you ever get a second derivative of y that is not zero?

[@carultch]

"Pure function" is likely a term he made up, so I don't think you'll find a definition in the textbook. What it means, is that it is ONLY a function of one of the variables, and not the other. The coefficients on y and all of its derivatives, can be functions of x and only x, as opposed to functions of both x & y. The derivative of y is a completely different issue than y raised to a power. The 2nd derivative of y is not the same thing as y^2. For a diffEQ to classify as linear, all of its y-terms can only be the original y-term, and any derivatives of y. The coefficients can either be constants, or functions of x and only x.

[@MuhammadImran-js5cs]

Thank you from us

[@atheerkhalaf6987]

thanks your video really helped me

[@worldwideCreator8032]

Thank you ,it was really helpful

[@nigelbryan7312]

Thanks...I'm going to pass my test in a few hours😃🏃

[@S1d2harth15]

brilliant and lucid explanation. thanks a lot!

[@TheMathSorcerer]

You are very welcome!

[@gianny2977]

thank you this was very helpful and to the point

[@abdulsamadfarooqui4585]

For part c, it should be non linear because it has a non linear term x^2?

[@GEMini_gurll]

Thank you so much but what do you mean by “pure functions”

[@asthakalet9741]

Awesome voice n explained in a well defined manner😊thnxx

[@TheMathSorcerer]

You are welcome😄

[@anyatisma5949]

This was amazing, thank you so much!!

[@Jack_lBlack]

Very helpful.. thank you!!

[@TheMathSorcerer]

You are welcome!

[@user-dx1rq8gl2n]

xy'+y(x+1)^2=sin(3x) is this equation linear or nonlinear ?

[@DoomKnight13]

Thank you so much. I know understand the difference.

[@oklol3641]

What if the equation has an e^(-x), instead of an e^(x), is that still linear?

[@TheMathSorcerer]

Yup

[@drewhaas2057]

Is the DE nonlinear every time you have a y on one side?? Where can I find example of these?

[@Al.Edelawit2749]

አመሠግናለሁ፡፡(Thank You So Much)

[@Spookbass]

Dude I love you. Thank you!

[@karangujarati4589]

Why don't we get teachers like you in our academics!??

[@2creativesisters266]

wow finally understood it! God bless you!

[@adhavankadhiravan4132]

Thank god, its very clear

[@TheMathSorcerer]

Glad it helped

[@martinjochnick8457]

Super helpful, thank you

[@casualworldchatcommunity5187]

Thank you!

[@TheMathSorcerer]

You are welcome!

[@_lu_ferro]

Really nice video, thanks!

[@oliveraherne5547]

+The Math Sorcerer Hi, thanks for the great video, loved your method but I'm wondering about a certain type of example. I.e. where you skip derivative terms. I know y''+y'+y=0 is linear but is y''+y=0 linear?

[@TheMathSorcerer]

yup both are linear!! Also both are super easy to solve, if you are bored and curious, see this video for a full explanation(including the derivation of the method) kzbin.info/www/bejne/q17cnqOnfZeGeNU

[@oliveraherne5547]

@@TheMathSorcerer Holy God, a maths lecturer making my life easier rather than harder. Thanks, you've got a new fan!

[@windthorpe9628]

y is not considered a pure function of x because it can be assumed to contain other variables other than x, right?

[@davidschori2417]

Does it matter if the DE skips a derivative order in the equation? example: y" + y = 0 or y" + y' = 0. Would this affect whether or not it is linear?

[@carultch]

Both of these are examples of linear differential equations. A diffEQ is considered linear, if there are only linear operations on the variable being differentiated (or any of its derivatives). By "linear operation", that means multiplying by a constant, adding a constant, or adding it in linearcombinations with other terms. When it "skips over" a derivative order, that just means there is an implied coefficient of zero on that particular term. It is still considered linear, even if there is a function of the independent variable multiplied out in front of it, like y" + sin(t)*y = 0. An example of a non-linear diffEQ is: y" + (y')^2 = 0 And it is non-linear, because the y' is being squared.

[@MShazarul]

Hi, I'm confused, if anyone can help out an ignorant. Example, for the highest order of derivative equals three, means the same as equal to derivative raised to the power of three?

[@gonzaloagullo5366]

Can anyone explain why the Y inside the cosine makes it nonlinear ?

[@hiddenblue1598]

Thank you sir, but may I know how 1 or any other constant is a pure function of x? Or we just assume, since any number can be a result of a certain function x? The video is great, can't totally get it in my book!

[@carultch]

"Pure function of x", means it is a function of x and no other variables that are relevant to the problem. So this means, the function in question, doesn't also depend on another variable like y, but instead, only depends on constants and x. It's true that constants could be due to some other variable in a more abstract sense, but that simply isn't within the scope of the problem in question. For instance, consider the linearized differential equation for a pendulum, which is L*d^2theta/dt^2 + g*theta = 0. The fact that g also depends on the bulk properties of Earth is irrelevant, since those don't change during the scope of the problem. Likewise, the length of the pendulum will also change with phenomena like thermal expansion, so it technically does depend on temperature. But again, that isn't relevant, since temperature rises are assumed to be negligible.

[@hiddenblue1598]

@@carultch Thanks sir!

[@ivankentcustodio9648]

Thanks bro!

[@saimwaqar4249]

Thanks for this

[@TheMathSorcerer]

You are welcome!

[@sammyomoto5456]

Two words, thank you.

[@aryanlehri3539]

In the example d u could have square both sides after taking y on right But then also it would have been non linear because of siny

[@rasivhengekhumbelo5370]

thank u very much

[@TheMathSorcerer]

You are welcome

[@queencathy7616]

Thanks i have understood

[@TheMathSorcerer]

Great!

[@hamadkhan2339]

Sorry to say but it is amazing...

[@TheMathSorcerer]

Thanks

[@raenraebautista4869]

Sir i need help can you tell me what is the linearity of this equation (dy/dx)= √(5x^2+1)

[@near1631]

Up

[@TheMathSorcerer]

Linear in y

[@raenraebautista4869]

@@TheMathSorcerer sir is it possible to change the linearity of a differential equations

[@raenraebautista4869]

if we raise both side by 2, it will be non-linear?

[@TheMathSorcerer]

Correct

[@euginemalama503]

Really helpful

[@TheMathSorcerer]

👍

[@tyrellaxe8841]

Thank you sorcerer guy

[@TheMathSorcerer]

You are welcome !

[@dhoom-z7221]

That really helped thanks

[@mohammadyousef6374]

THANK YOU ALOOOT , KEEP UP THE GOOD WORK SIR!!

[@serkandonmez1536]

you are the best bro

[@itzyryeji3164]

what do you mean a pure function of x? it’s the part that i can’t understand. thank you!

[@TheMathSorcerer]

Just a function of x...no other variables

[@orions7595]

@@TheMathSorcerer so a constant is a function of x?

[@Degenerategenerator]

thank you.

[@bozosmith3506]

Why is it important to make this classification? What's the difference?

[@TheMathSorcerer]

Linear DE's can be solved, all of them, whereas nonlinear DEs are harder.

[@bozosmith3506]

@@TheMathSorcerer Thanks. But now, with fast computers and iterative solving, this has become kind of irrelevent. Right?

[@TheMathSorcerer]

@@bozosmith3506 I really don't think so , I do think it's important to know how to solve DE's by hand and of course to be able to tell a linear one from a nonlinear one. I think that a lot of times people think that because computers can do stuff we no longer need to be able to do things by hand. This is a common point of view I think.

[@haydenford38]

Would this process also work with partial derivatives? Such as F(x)*F(y)=xy

[@omersuveyd5503]

Thank You, this helped me a lot :)

[@TheMathSorcerer]

Awesome!

[@shihabuddin7632]

You saved me!

[@raadelmsellati8438]

I love you brother

[@TheMathSorcerer]

❤️

[@abpdev]

Thank you for this one bro!