Separable Differential Equations (Differential Equations 12)

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For real bro

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real af

[@ThatDereKid]

"...Partial fractions. Let's take a moment to review that." THANK YOU! I cannot thank you enough for reviewing things. So many teachers are prideful and pretend that students shouldn't have to review or that it isn't worth class time. I'm taking Diff EQ this semester at a top 50 university and that is what my teacher did about partial fractions. Meanwhile you walk through an example that is properly explained and point students toward a full Calc 2 video (that you MADE) in case the singled out examples here makes them realize that they really need to review the concept in full. Your teaching is proof that many others who teach with a false style of "practical arrogance" simply have no excuse. Thanks.

[@ProfessorLeonard]

You're so welcome! I think it's an absolute shame that many professors consider arrogance to be acceptable in academia. It's VERY hard to create a quality learning atmosphere in such conditions. Best of luck in Diff Eq!!

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Professor Leonard said, "Im proud of you, you're solving differential equations, that's pretty cool". Thank you professor Leonard, you have helped me through my time at community college and have helped me transfer to a 4 year university, Thank you!

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"you're solving differential equations, pretty cool. I'm proud of you , man" My life force returned at this very moment

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If you want to get buff in general then focus on compound exercises, such as Squats, Deadlift, Military Press, Bench Press, Pullups, Walking Lunges and Farmer's Walk. Basically pick 3-4 of those exercises each day and you can't go wrong.

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And if you want to focus some extra on the biceps, then Hammer Curls and Concentration Curls might be worth a try. But you should also focus a lot on the triceps, since it actually takes up 2/3 of the arm, and for that I recommend Cross-Bench Tricep Dips. But both of these muscle groups will get a lot of training from the aforementioned compound exercises as well.

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gradient operator crossed with F

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@@dal-qi3gv the equation for a concentrated curl is sqrt(50x) on the interval [0,100]

[@hourz]

Man its crazy to me I was watching his pre algebra course in 2020 at 30 years old with basically a 4th grade math level which was stated on my records on my HS graduation 12 years ago. now I'm in calculus 2 about to graduate from my community college after this fall semester. I hope I don't hit the celling with calculus 2. however, what I learned about myself I am not afraid of failure and persist until I make it. Thank you professor Leonard!

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We as students could never thank you enough for what remarkable contribution you have in building our concepts firmly.You have definitely helped hundreds if not thousands of students. I owe you my academic success prof leonard!!!!

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I finally passed calculus 2 now i can watch this 🤣💔

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That's a whole mood right there

[@4seth]

I am taking this in calc II :(

[@sydneybrown7473]

Thank you for always explaining the steps in integration, had a problem with partial fractions in lecture today and the professor didn't go through the steps at all and no one remembered how to do them. Your videos saved me in multivariable calculus, and I look forward to learning with you in differential equations as well!

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dude. You have no idea how USEFUL your videos are to me!!! I wish i had discovered you before my exams...

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RIP

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I'm currently in community college studying to get my associates to transfer to a 4 year school for an engineering degree. This wouldn't be noteworthy unless I also mention the fact that math was consistently my worst subject growing up, to the point where math was the only class that I ever failed in high school. With that being said I got an A on my last Calc 2 exam using your videos and your explanations, passion, and encouragement are all very large parts in my success on that test. I'm writing this comment so I can come back in a couple of years and link a picture of my engineering ring that will be mine in no small part thanks to you :) thank you!

[@frustrated3880]

It might be late or maybe some people have already commented this, but you can actually *set the arbitrary constant 'c'* as *ln|c|* and combine it with the term that uses 'ln', too.. for example, in the 1st problem, ln|y|=-x^2 + c1 can be written as ln|y| = -x^2 + ln|c1| , then ln|y|-ln|c| = -x^2, to ln|y/c| = -x^2. Putting 'e' on both sides will result into e^ln|y/c| or y/c = e^(-x^2). You will then finish with the solution which is *y=ce^(-x^2)* without going through an additional process..

[@bmoreno-ri8]

Why's that legal? I mean why c = ln|c| ?

[@frustrated3880]

@@bmoreno-ri8 because C is an ARBITRARY constant. Meaning it can be anything you set it to be. The natural log of a constant is still a constant, so it’s valid.

[@bmoreno-ri8]

@@frustrated3880 so I can also do things like a^c and that's still a constant and it's legal?

[@frustrated3880]

@@bmoreno-ri8 yeah, and you can set it as stuff like sin(c) as well and you’ll be fine :)

[@actuary32174]

Are you planning on covering partial differential equations in this course? Thank you for all of your efforts.

[@HS-yx5by]

You are seriously my saving grace. Thank you so much for these! The only reason I passed calculus 3 and now calculus 4 is because of you. Wishing you all the best!

[@joshuaingya4418]

Sir, you are my best maths tutor ever. I have understood all you with no stress at all. Fortunately, I'm currently being taught differential equations in the university at the moment. Thank you sir.

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Thank you, Professor. I think other people already said how much we appreciate these lectures.

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the pep talk he gave somewhere in here brought the sparkle back into my eye. thanks professor.

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he's got 340K now:)

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Now he's got 525k subs :)

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@@NemoTheGlover and now at 564k....perfect

[@Rick_Sanchez_Jr.]

Thank god for Professor Leonard, we just started these topics in our Calc 2 class and it barely makes sense whatsoever, Professor Leonard explains it so much better. Thank you.

[@goldengeek3320]

@@kyle1538 Standard calc 2 stuff... separable differential equations are about as simple as they come. I'm watching this for calc 2 as well. I was too lazy to shuffle through his calc 2 videos.

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This is truly gold, i'm studying it for my coming spring semester, you are the best teacher ever!!!

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Super exciting time at this end because the semester is about to be over, yet I want to iron out some last minute crinkles. Your videos are nothing short of miracles! Thank you professor!

[@DaHoKilla]

Leonard you are the goat im so glad to have found your channel, saved me in calc 2,3, and now DE!!

[@christopherbarnes6976]

Thank you for taking the time to show the integration techniques

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Woooo glad I remembered about this channel. I’m not in school but I love watching these videos. Thanks so much

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I dreamt of the day he'd post this video. Even though I took Diffy q's already, I will Definitely watch these as refreshers.

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Best lesson I have had. Thanks Prof

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I greatly appreciate your video's. I've been watching since Calc 1.

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Thank you; it's really difficult to find good material on this, and this couldn't be better.

[@zomnom8289]

Thank you Professor Leonard. I didn't even know I was doing something wrong in my Dif EQ class until now because my teacher doesn't tell us what we did wrong on quizzes or exams. Thanks so much.

[@iancoleman6352]

My professor cares about math and making sure students understand the material but, especially with online classes, it is much more difficult to interact and the presentation quality of the class goes down. This translates to a lot of frustration and confusion on the students' end. Your video presentation is excellent and I am greatly appreciative of the massive amount of examples used in the videos and the thorough explanation of each one. Seriously, thank you. You make mathematics accessible to a larger audience.

[@jazminwalls2591]

I can not THANK YOU ENOUGH for going over step by step. Unfortunately I did not find you until the 3 days before my final was due so I will be completely honest, I struggle with integrating. I'm taking calculus 3 and differential equations at the same time and wow it is a lot. But you actually give me some hope of passing. Thanks again!!!

[@aznperswazinable]

My dude, you make understanding Diff EQ so well !

[@metalmanny0000]

Thank you so much professor. I'm taking Diff Eq. this semester and not going to lie i have been lost since the beginning of class,and my professor is difficult to understand(language barrier). After watching this video( and doing the practice problems before you worked them out) i feel really confident about this section and will do so with the other cases. I have no doubt that i will be ready for my first exam next week thanks to your awesome lecturing videos.I hope to see linear equations soon, will be taking that in the fall.

[@neondino9790]

My professor didn't really explain this well for me to understand but after watching this I have a better understanding of Separation! Thank you!!

[@noel2577]

1:20:08 I did that on my own , but Leonard's reaction had me dying!!! thanks

[@thegreatbritishbikeoff]

1:19:50 that look when you realize you’ve led you students into the abyss. Love the vids professor. Your love for your craft is inspiring!

[@LightCastells]

I was so mad....

[@miqomargaryan15]

There is one thing that Leonard didn't mention here. We cant replace e to the power of C_1 by C all the time because e to the power of any constant can't be zero, so we need to exclude the case when C is equal to zero and then check whether 0 is a solution to the differential equation and if it is so, then we can write C at the place of C_1. Fortunately, without applying this technique, the problems are solved correctly. However, it was an important part to talk about.

[@auradkamkar627]

This is where the fun stuff happens so pumped for this lecture!

[@JITCompilation]

Thank for teaching me about differential equations, Stephen Colbert

[@Shadowbrothe]

When you asked if that made sense, I raised my hand like I do in all your other videos!

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Thanks for this videos Professor, i can get it now

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u are such a magnificent proff that I can't enjoy maths without watching ur videos I pray forMore blessings for u

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Thank you so much for all these amazingly helpful videos!

[@snah70]

It's a good thing that you're careful about handling the absolute values correctly, but you should also worry about not dividing by zero! For instance, in the very first example, dy/dx'=-2xy holds if and only if dy/y=-2x dx OR y(x) is identically zero. In your computation, C = plus/minus exp(C_1) is an arbitrary NONZERO constant (since C_1 is an arbitrary real constant, and the sign is arbitrary), so you don't actually get the whole general solution, where C=0 is also allowed; this case corresponds to the exceptional solution y(x)=0, and needs to be added in “by hand” at the end.

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You are an amazing teacher!!! You make everything so easy to understand. Thank you. You also look like Superman btw, what a math hero.

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Missed my math class on this topic cause I was sick. Thanks for covering this!

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YOU SAVE ME EVERY TIME. THANK YOU

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The Man! Thank you professor...

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GOOD WORK

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Excellent Series, thank you for your generosity.

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Thank you Professor, just had the lightbulb moment

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kosovo isn't a country

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Thanks a lot bro I really appreciate your videos I'm having trouble right now in calculus 4, I hope that before next semester you can get all the way through matrices and things like that a vector spaces find Subspace and all that

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You are the best !!!!!! This really helped me ..

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thank you life saver during this online class thing.

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i was getting frustrated because I didn't understand but damn... you helped me a lot. Thank you so so so much, ill watch your videos from now on.

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You are the reason of my better grades. What do I even pay my university for?

[@theformerknights8538]

whether or not on purpose, i will choose to believe the small mistakes he makes and catches are actually done on purpose as they are useful tools for my own learning. I've been stoked that I've caught some 'before' he has. But this leads me to believe it all his plan anyway. I hope you'll be back for more lectures. my current professor could NOT care less about teaching effectively.

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Smart, handsome, and fit professor. Very in depth videos

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thanks professor leonard for everything!

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Thank you very much!!

[@AB-gu9ui]

@1:21:59 You could also use a simple u sub to find the integral of tan(y)

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God bless you Professor Leonard! You are just an incredible teacher. May Jesus add more wisdoms and knowledge to you!

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Honestly, only reason Ill graduate financial math uni is because of this legend!

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saving me right now. Maaaan. Thank you!

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the best teacher here

[@dragosmanailoiu9544]

Great video prof amazing examples at the end may I request harder integrals please

[@ti6252]

this video was a great help keep going prof

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Professor Leonard, will you be looking at PDE's, dirac delta, and fourier series, with this series?

[@johnkiserliss123]

In the example 2XDY/DX+Y=10sqrx the integrating factor is quite obvious. As it is just looking at you.

[@coleabrahams9331]

16:13 it's called exponentiating. We are exponentiating both sides of the equation.

[@NovaWarrior77]

Am in advanced classical physics. Had to pay this bad boy a visit again.

[@Adolf0is0winner]

Professor Leonard, you are an excellent teacher and excellent at explaining but it seems that you have made mistake in adding the fractions , no worries as I make tonnes of mistakes on daily basis . Thanks for teach the differential equations . I learnt a lot from you , Jason and organic chemistry tutor.

[@Bee-uy2cn]

“Were gonna take e -well not gonna take e..hehe” 16:14 i caught that 😂

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literally stopped the video and came down to see if anyone had said anything about that LOL

[@Micholin16]

I just noticed it and laughed harder than i should

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[@Peter_1986]

Yeah, he has the Overall Quality rating 5.0 on "Rate My Professors", and almost everyone has rated him "Awesome", lol.

[@Mahmoudalkasem1997]

great video thanks for your efforts

[@hitm43]

36:05 The notation on the fourth power in the exponent of the natural log makes it look like the ENTIRE natural log is being raised to the fourth power which is not the case. The argument |1+x| should be raised to the fourth power, not the entire natural log.

[@lovesandwich8783]

The math teacher we all wanted.

[@zaynbashtash]

Absolute legend

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I love this guy

[@moeberry8226]

The reason you don’t get y=0 as a solution is because when solving seperable differential equations you are dividing by y in this case but there’s the restriction before integrating that y cannot be 0. And then you handle that case separately at the end or even at the beginning, what ever you choose.

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I know this your channel is for mostly math but what is your workout routine?

[@DanielPerez-mb9hk]

Thank you for these videos they are super helpful!

[@ahmedaltaleb2930]

Hello Professor Leonard thank you for the great video I solved the problem before last in a different way and I get this answer y=3/(3x+x^3)+c I was wondering how could I check my answer or what the mistake I make my first manipulation was this x^2 -1-x^2= y^2 -x^2 y^2 is removing the X^2 a wrong move if so where could I found further sources to explain this type of mistake

[@gianny2977]

At 59:20 don't we have to include 2 and then its outside of the natural log so we put it up as an exponent to this fraction of (1-x)/(1+x)? Where does it go? and Thank you so much!!!

[@sshilovsky]

You missed the solution y(x)=0 in the second example: `dy/dx + 2xy^2=0`. When splitting x's and y's, you divided by y, implicitly assuming it's not zero in any point. Same problem exists in the first example, but the eventual solution was correct. Another problem in the first example though is that +/- is not necessarily a const, when you just hide it into `C` (it actually is a const, ofc, but it has to be demonstrated). Alternatively, instead of checking these steps strictly, we can verify the resulting solution set, and then see if it covers the whole domain of f(x,y). It works for example #1, but not for example #2. Sorry for poor English, I hope it's decipherable.

[@nurgulquliyeva7296]

I laughed so hard to that face after saying "sorry if you did that on your own" and suddenly you get the serious face saying "good job". You are smart, handsome and funny. Oh lord, such a superman!

[@namehere630]

thank you for this

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i like this. my confusion from class is healing

[@QueenL]

it would be awesome if you do videos of calculus IV, you explained clearly, thanks.

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if you were my professor id def become a math major