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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@@ProfessorLeonard dude i wish you were my teacher

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Sometimes I have to watch his video at least twice because of his biceps First time: His muscle.. Second time: very detailed explanation.. Got it!!

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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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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]

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I really enjoy watching the works of Professor Leonard. He is indeed a true mathematics teacher who takes his time to present his content. I admire him so much. Thank You Prof.

[@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!!!!

[@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!

[@ethan5306]

Great video, definitely appreciate the focus on algebra in this video. The bane of Calculus so far has definitely been the algebra

[@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!

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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!

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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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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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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 :(

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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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Let's all share his channel and get the Prof. To 300k subs by the end of the week!!! He's almost there. Hit up all the math blogs!!!!

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he's got 340K now:)

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Now he got 487k subscribers;-)

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Now he's got 525k subs :)

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now 560k

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

[@giabachng]

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

[@actuary32174]

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

[@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 :)

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Deca

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[@christopherbarnes6976]

Thank you for taking the time to show the integration techniques

[@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.

[@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!

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I was so mad....

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

[@DaHoKilla]

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

[@accensa]

11:17 35:47 (logarithm property) 1:04:20 (need one fraction to reciprocate, consider common denominator) 1:25:41 (factor techniques)

[@accensa]

1:31:39 (singular solutions)

[@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.

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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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[@ddahstan6876]

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!

[@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.

[@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!!

[@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!!!

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

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you are more than amazing, better than any math teacher i ever seen

[@Shadowbrothe]

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

[@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.

[@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.

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My dude, you make understanding Diff EQ so well !

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

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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 for teaching me about differential equations, Stephen Colbert

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Best lesson I have had. Thanks Prof

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

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

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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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Thank you so much for all these amazingly helpful videos!

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

[@noel2577]

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

[@coleabrahams9331]

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

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

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Thanks for this videos Professor, i can get it now

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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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Excellent Series, thank you for your generosity.

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I love you man, I hope for good fortune to you forever!

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thank you life saver during this online class thing.

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Thank you Professor, just had the lightbulb moment

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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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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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The math teacher we all wanted.

[@AB-gu9ui]

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

[@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.

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I would be grateful if you could make a series on "Tensor calculus".

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kosovo isn't a country

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the best teacher here

[@johnkiserliss123]

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

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YOU SAVE ME EVERY TIME. THANK YOU

[@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.

[@chemicalengineeringbd7614]

Quran : surah Al-mulk تَبَارَكَ الَّذِي بِيَدِهِ الْمُلْكُ وَهُوَ عَلَى كُلِّ شَيْءٍ قَدِيرٌ Blessed is He in whose hand is dominion, and He is over all things competent الَّذِي خَلَقَ الْمَوْتَ وَالْحَيَاةَ لِيَبْلُوَكُمْ أَيُّكُمْ أَحْسَنُ عَمَلًا ۚ وَهُوَ الْعَزِيزُ الْغَفُورُ [He] who created death and life to test you [as to] which of you is best in deed - and He is the Exalted in Might, the Forgiving الَّذِي خَلَقَ سَبْعَ سَمَاوَاتٍ طِبَاقًا مَا تَرَى فِي خَلْقِ الرَّحْمَنِ مِنْ تَفَاوُتٍ فَارْجِعِ الْبَصَرَ هَلْ تَرَى مِنْ فُطُورٍ [And] who created seven heavens in layers. You do not see in the creation of the Most Merciful any inconsistency. So return [your] vision [to the sky]; do you see any breaks? ثُمَّ ارْجِعِ الْبَصَرَ كَرَّتَيْنِ يَنْقَلِبْ إِلَيْكَ الْبَصَرُ خَاسِئًا وَهُوَ حَسِيرٌ Then return [your] vision twice again. [Your] vision will return to you humbled while it is fatigued. وَلَقَدْ زَيَّنَّا السَّمَاءَ الدُّنْيَا بِمَصَابِيحَ وَجَعَلْنَاهَا رُجُومًا لِلشَّيَاطِينِ وَأَعْتَدْنَا لَهُمْ عَذَابَ السَّعِيرِ And We have certainly beautified the nearest heaven with stars and have made [from] them what is thrown at the devils and have prepared for them the punishment of the Blaze. وَلِلَّذِينَ كَفَرُوا بِرَبِّهِمْ عَذَابُ جَهَنَّمَ وَبِئْسَ الْمَصِيرُ And for those who disbelieved in their Lord is the punishment of Hell, and wretched is the destination. إِذَا أُلْقُوا فِيهَا سَمِعُوا لَهَا شَهِيقًا وَهِيَ تَفُورُ When they are thrown into it, they hear from it a [dreadful] inhaling while it boils up. تَكَادُ تَمَيَّزُ مِنَ الْغَيْظِ كُلَّمَا أُلْقِيَ فِيهَا فَوْجٌ سَأَلَهُمْ خَزَنَتُهَا أَلَمْ يَأْتِكُمْ نَذِيرٌ It almost bursts with rage. Every time a company is thrown into it, its keepers ask them, "Did there not come to you a warner?" قَالُوا بَلَى قَدْ جَاءَنَا نَذِيرٌ فَكَذَّبْنَا وَقُلْنَا مَا نَزَّلَ اللَّهُ مِنْ شَيْءٍ إِنْ أَنْتُمْ إِلَّا فِي ضَلَالٍ كَبِيرٍ They will say," Yes, a warner had come to us, but we denied and said, ' Allah has not sent down anything. You are not but in great error.' " وَقَالُوا لَوْ كُنَّا نَسْمَعُ أَوْ نَعْقِلُ مَا كُنَّا فِي أَصْحَابِ السَّعِيرِ And they will say, "If only we had been listening or reasoning, we would not be among the companions of the Blaze." فَاعْتَرَفُوا بِذَنْبِهِمْ فَسُحْقًا لِأَصْحَابِ السَّعِيرِ And they will admit their sin, so [it is] alienation for the companions of the Blaze. إِنَّ الَّذِينَ يَخْشَوْنَ رَبَّهُمْ بِالْغَيْبِ لَهُمْ مَغْفِرَةٌ وَأَجْرٌ كَبِيرٌ Indeed, those who fear their Lord unseen will have forgiveness and great reward. وَأَسِرُّوا قَوْلَكُمْ أَوِ اجْهَرُوا بِهِ إِنَّهُ عَلِيمٌ بِذَاتِ الصُّدُورِ And conceal your speech or publicize it; indeed, He is Knowing of that within the breasts. أَلَا يَعْلَمُ مَنْ خَلَقَ وَهُوَ اللَّطِيفُ الْخَبِيرُ Does He who created not know, while He is the Subtle, the Acquainted? هُوَ الَّذِي جَعَلَ لَكُمُ الْأَرْضَ ذَلُولًا فَامْشُوا فِي مَنَاكِبِهَا وَكُلُوا مِنْ رِزْقِهِ وَإِلَيْهِ النُّشُورُ It is He who made the earth tame for you - so walk among its slopes and eat of His provision - and to Him is the resurrection. أَأَمِنْتُمْ مَنْ فِي السَّمَاءِ أَنْ يَخْسِفَ بِكُمُ الْأَرْضَ فَإِذَا هِيَ تَمُورُ Do you feel secure that He who [holds authority] in the heaven would not cause the earth to swallow you and suddenly it would sway? أَمْ أَمِنْتُمْ مَنْ فِي السَّمَاءِ أَنْ يُرْسِلَ عَلَيْكُمْ حَاصِبًا فَسَتَعْلَمُونَ كَيْفَ نَذِيرِ Or do you feel secure that He who [holds authority] in the heaven would not send against you a storm of stones? Then you would know how [severe] was My warning. وَلَقَدْ كَذَّبَ الَّذِينَ مِنْ قَبْلِهِمْ فَكَيْفَ كَانَ نَكِيرِ And already had those before them denied, and how [terrible] was My reproach. أَوَلَمْ يَرَوْا إِلَى الطَّيْرِ فَوْقَهُمْ صَافَّاتٍ وَيَقْبِضْنَ مَا يُمْسِكُهُنَّ إِلَّا الرَّحْمَنُ إِنَّهُ بِكُلِّ شَيْءٍ بَصِيرٌ Do they not see the birds above them with wings outspread and [sometimes] folded in? None holds them [aloft] except the Most Merciful. Indeed He is, of all things, Seeing. أَمَّنْ هَذَا الَّذِي هُوَ جُنْدٌ لَكُمْ يَنْصُرُكُمْ مِنْ دُونِ الرَّحْمَنِ إِنِ الْكَافِرُونَ إِلَّا فِي غُرُورٍ Or who is it that could be an army for you to aid you other than the Most Merciful? The disbelievers are not but in delusion. أَمَّنْ هَذَا الَّذِي يَرْزُقُكُمْ إِنْ أَمْسَكَ رِزْقَهُ بَلْ لَجُّوا فِي عُتُوٍّ وَنُفُورٍ Or who is it that could provide for you if He withheld His provision? But they have persisted in insolence and aversion. أَفَمَنْ يَمْشِي مُكِبًّا عَلَى وَجْهِهِ أَهْدَى أَمَّنْ يَمْشِي سَوِيًّا عَلَى صِرَاطٍ مُسْتَقِيمٍ Then is one who walks fallen on his face better guided or one who walks erect on a straight path? قُلْ هُوَ الَّذِي أَنْشَأَكُمْ وَجَعَلَ لَكُمُ السَّمْعَ وَالْأَبْصَارَ وَالْأَفْئِدَةَ قَلِيلًا مَا تَشْكُرُونَ Say, "It is He who has produced you and made for you hearing and vision and hearts; little are you grateful." قُلْ هُوَ الَّذِي ذَرَأَكُمْ فِي الْأَرْضِ وَإِلَيْهِ تُحْشَرُونَ Say, " It is He who has multiplied you throughout the earth, and to Him you will be gathered." وَيَقُولُونَ مَتَى هَذَا الْوَعْدُ إِنْ كُنْتُمْ صَادِقِينَ And they say, "When is this promise, if you should be truthful?" قُلْ إِنَّمَا الْعِلْمُ عِنْدَ اللَّهِ وَإِنَّمَا أَنَا نَذِيرٌ مُبِينٌ Say, "The knowledge is only with Allah, and I am only a clear warner." فَلَمَّا رَأَوْهُ زُلْفَةً سِيئَتْ وُجُوهُ الَّذِينَ كَفَرُوا وَقِيلَ هَذَا الَّذِي كُنْتُمْ بِهِ تَدَّعُونَ But when they see it approaching, the faces of those who disbelieve will be distressed, and it will be said, "This is that for which you used to call." قُلْ أَرَأَيْتُمْ إِنْ أَهْلَكَنِيَ اللَّهُ وَمَنْ مَعِيَ أَوْ رَحِمَنَا فَمَنْ يُجِيرُ الْكَافِرِينَ مِنْ عَذَابٍ أَلِيمٍ Say, [O Muhammad], "Have you considered: whether Allah should cause my death and those with me or have mercy upon us, who can protect the disbelievers from a painful punishment?" قُلْ هُوَ الرَّحْمَنُ آمَنَّا بِهِ وَعَلَيْهِ تَوَكَّلْنَا فَسَتَعْلَمُونَ مَنْ هُوَ فِي ضَلَالٍ مُبِينٍ Say, "He is the Most Merciful; we have believed in Him, and upon Him we have relied. And you will [come to] know who it is that is in clear error." قُلْ أَرَأَيْتُمْ إِنْ أَصْبَحَ مَاؤُكُمْ غَوْرًا فَمَنْ يَأْتِيكُمْ بِمَاءٍ مَعِينٍ Say, "Have you considered: if your water was to become sunken [into the earth], then who could bring you flowing water?" Thank you for the lesson. That was my gift to you translated surah

[@Bee-uy2cn]

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

[@julianfernandez5510]

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

[@mentor1324]

thanks professor leonard for everything!

[@atpTUBE]

The Man! Thank you professor...

[@umerwaheed1972]

Fourth content creator, for whom I actually felt that I need to donate to on Patreon: 1. GanjIsSwag 2. Mooroo 3. CinnamonToastKen 4. Professor Leonard

[@cynthianwosu3237]

Thank you so much for all of ya videos..is been so helpful.🥰

[@kaneelsnuiver]

I'm throwing away my book, thanks for these vids.

[@nunoskid153]

if you were my professor id def become a math major

[@user-fk5di4me9c]

i fucking love this guy why tf i haven't watched his videos from the beginning

[@ahmedalzaabi5848]

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

[@abrhamaddisu-ht4qp]

GOOD WORK

[@scorpionesp6510]

saving me right now. Maaaan. Thank you!

[@ti6252]

this video was a great help keep going prof

[@advisualz5630]

i just want to say thank you for making math understandable

[@vjosaveselaj3788]

Thank you very much!!

[@dragosmanailoiu9544]

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

[@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.

[@zaynbashtash]

Absolute legend

[@taekwondotime]

*I wish he would have done more examples where the terms weren't products* . The first thing I ran into when I began doing exercise questions in my textbook was questions where you have "+" or "-" some term. I would try to integrate and couldn't figure out what I was doing wrong. The fact that you need *PRODUCT TERMS* before integrating is probably the #1 thing that will bury students when doing separable differential equations. It's as though the integrals you've been doing all your life suddenly don't work with separable DEs. *Separable DE Exercise Questions from my textbook:* 3. y' = 1 + 0.02y^2 (Good luck! That was the opening question and it buried me.) 5. y' = xy/2 11. y' + csc(y) = 0 13. xy' + y = 0 ; y(2) = -2 15. e^xy' = 2(x+1)y^2 ; y(0) = 1/6 17. y' cosh^2(x) - sin^2(y) = 0 ; y(0) = Pi/2 If you can solve the questions above, then you can say you truly understand this section.

[@ProfessorLeonard]

So, remember that if you need products and you have + or - , you'll either be adding/subtracting or factoring or using some identity. Also, keep in mind that an expression with more than one term can be it's own factor. A lot of these separables that we run across are the type where we add to the other side and then divide.... hey that rhymes!

[@darkmatter5631]

That left bicep is actually nuts!

[@jeremybarber2158]

this guys a great teacher

[@Peter_1986]

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

[@elipayne2617]

You are a legend, thank you.

[@siyandamaseko2312]

I love this guy