This is why you're learning differential equations 

When I teach about differential equations, I'll make sure to do a similar introduction, it's really brilliant

And God said let there be dx

When I learned physics at University one of the most satisfying experiences was getting to grips with calculus enough to use it to derive equations as it allowed me to wield mathematics to describe the world, which felt awesome.

I asked my Math teacher the same question when he was teaching DE. And the whole class laughed at me. Thanks for the video.

them: do you even lift bro? me: yes. barbells, with no mass.

Me wanting to be an engineer: Haha, I'm in danger

"Let's assume there's no wind" When have I seen this line before..

One of the best videos ive seen, it gave me some sort pf motivation to finally pay attention in math class since my teacher never explains why we are learning what we are learning

Differential Equations are definitely a household name in the applications of maths. They are really fun when you know how to utilize all of their techniques. Awesome work!

Good to know Zach :D

When I studied aeronautical engineering at university I was amazed when I discovered that DE are the key to model many physical problems no matter if it is structural mechanics, thermodynamics, aerodynamics or electrical engineering.

Me acting as if I understood anything he just said: 🤓🤓

Had a similar chain problem on a physics exam. Still haven't recovered mentally/emotionally.

"Differential equations are cool" ~Big Bang

I am learning differential to slap my brother perfectly at the moment when he would be running to tell my mom that I failed in math.

You know, I really enjoyed calculus last year and understood many of the real life applications, but I never knew about the pursuit curve thing before. Mathematics is just something else man.

Me watching this, avoiding actually learning the differential equations.

People who like math: wow this is interesting Me, who barely passed calc 2 and linear Algebra: *screeching noises*

It's wierd that in schools we are taught to solve them by just learning the formulas and getting the right answers and not the practical usage. It would have been easier to learn about them by having them connected to some practical usage.

I love how intuitive he is with explaining it, it's easier to understand why it's important and gives it meaning.. something few ppl do but it seperates the good from not so good at explaining things

4:10 There's this famous example of a pursuit curve. The French wanted to make an anti ship missile. So they built a missile, slapped on a sensor, and then programmed the missile to always fly towards the target. In the first test, they launched against a *small* "drone" target. As the distance approached 0, the tightness of the turn required to remain on target approaches infinity. So it missed. It now locked on to a stationary buoy. But the buoy was WAY too *small* for the missile to hit, so it missed. Finally, it locked on to a *large* civilian cargo ship, which was minding its own business. The testers panic, because that ship is in range, it's so large they think the missile WILL hit, and they didn't bother to install a remote termination system. The missile again lowers its altitude to the sea skimming attack pattern. It makes the last course adjustments... ...and misses. Finally it runs out of fuel and crashes into a busy shipping lane. When Sweden's FOA (defence research institution) learnt of this, they fed the missile's guidance algorithm into their brand new (for that time) supercomputer, and the output predicted the exact flight pattern of that French missile. That was when the French leant about "CBDR-pursuit" (Constant Bearing Decreasing Range) and "PIP-guidance" (Predicted Impact Point).

gravity rounded to 10 * Laughs in physics *

Me: Why are we learning this? Teacher: So, that's how you're going to solve it when it appears in examination Me: 🙃

Just like how grass is always greener on the other side, everybody likes to say that their education systems are bad etc. But I'm so glad that our state school syllabus (Indian) had a chapter in 12th Maths called "Application of derivatives" and it taught us practical applications of differential equations. It had problems very similar to those shown in the video. Motion of a rocket with time varying mass, filling up of conical containers with time varying radius, maximizing volume of solids for given surface areas, etc. And I must say, that is what really made me fall in love with differential equations. I just laugh at people who say "When am I ever going to use calculus in real life?" Because I know how useful they really are.

Damn I wish more people understood just how mind blowing physics and Calculus really are. Like these equations describe the damn universe, the universe itself runs on mathematics and physics. Mathematics is literally God's language and it's so beautiful.

"round gravity to 10 as always" haha

Former engineer, now a nurse, no matter how much I gorge my brain on new medical terminology, biology and pharmacology, I always find myself coming back to the topic of STEM. I wish I could go back to engineering but the jobs just aren’t there, everyone only wants to hire for short term projects.

I feel like if I knew the application of differential equations, I would have enjoyed that course much more. This video makes me want to revisit those concepts. Thanks.

DEs was always one of my favorite courses

Mann I was just solving differential equations when this video popped up!

Knowing why I will be taking Diff Eq. this coming semester is half the battle. Everything you covered was very interesting to me and now I'm looking forward to taking the class and expanding my knowledge. Thanks for the informative video!

Did my master's thesis on a differential equation, loved your breakdown!

Wondering video illustrating not simply the awesome power of calculus, but the concept as well. The conceptual thinking here is key as this is how we begin to develop models for analysis!

It has been many decades since I first learned various methods for solving differential equations, and used them throughout my career as an engineer. I remain impressed about how easy it is to write down a differential equation, as opposed to the difficulty in solving it. Despite advanced methods of numerical analyses, analytical solutions still come down to educated guesswork: intuit a possible solution, plug it into the equation, and find out if it works.

Im in my last Calculus 3 class taking Diff Equations next semester and this looks pretty interesting, I can't wait! Thank you for the vid Zach!

@Zach Star Thank you for presenting these awesome real-world applications of differential equations! It really helps to understand how to apply the mathematics that we learn or are going to learn. Please keep the educational and informative videos coming. I really appreciate what you are doing.

"I don't care I'm going to economics" Macro & interest growth differential equations(and friends): *"allow us to introduce ourselves"* Edit: highlighted the irony more clearly

Great video, I took differential equations as an undergrad pre-requisite to mechanical engineering at UT. We called it 'difficult equations', but made sense when I later got into applications (i.e. state space model diff. eqs.).

I wish i had this RU-vid when i went to school. People like you make the most valuable part of it, amazing. Thank you!

wow i am feeling great to have found about a channel dedicated to application rather than only to theorem which we can easily find in textbook. KEEP IT UP!!!

How do I get girls to like me? Differential equations How do I get this stain out of my pants? Differential equations What happens if I can't solve a differential equation? Try different differential equations That's right folks they do it all. They can slice and they can dice, make your teeth whiter, and your car go faster. and all for the low low... ok im done.

- This was way over my head but I enjoyed how you presented it. The equation for constant change in mass as the shuttle rises instead of Newtons really struck me to how complicated a launch is. 🤔 👍

Out of all my math classes. Diff EQ is my favourite. It was fun figuring things out and having it all come together

I wish i had a maths teacher like him Instead of just blindly solving the equations by formula, we know how to imagine & understand

Zach, Just brilliant. I am always curious about the implications of the math I am learning. Thanks a lot

I consider Differential eqn as a branch of mathematics (actually Calculus) which are very intresting and very helpful in every day life.Btw thx for this video! Good luck!

Thanks for making me dread this next semester a little bit less haha. Love you’re comedy videos but I think I love the teaching ones even more!

You have a teaching gift, thanks for putting this together. Subbed.

I learned these a year ago and went through hell to pass the class and I’ve forgotten them all.

During my time at school... Something which you can't perform outright seemed boring...but now watching yt videos getting context of what is the actual application of these equation is really fascinating...Applications were there in the textbooks but we're not at all relatable as those were some mumbo jumbo high level experiments...

For pursuit curves, if you do not know the trajectory of the pursued object/person/variable, you can use a Kalman filter or polynomial regression to predict it, then apply the pursuit algorithm...

I am just gonna start teaching professionally and this is pure gold to make students interested. Thank you so much.

Consider doing a episode on Integral Equations. These seem to get overlooked in the typical "Math Methods" course, but then are used extensively in Graduate Level Physics, "Jackson", etc.

Haha, Zach you just had to round up gravity. LMAO!

Thanks for such a wonderful and awesome set induction about differential equation.

I just finished algebra ii, and yet I still completely understand the first ten minutes (except for some of the math around 8 minutes) of the video. So brilliantly explained! Thanks.

Just finished calc 3 and linear algebra and I can finally understand math that they do in RU-vid videos

Zach star and 3B1B . Perfect!

I guess one brilliant description for this amazing teaching/explanation is BRILLIANT!

Fascinating video. Thanks for your hard work!

Man if I had the same passion watching this video when I barely passed Maths.

Nobody: Me: I have differential equations test in 2 hours, I think I'll watch this video right now.

I was reading about Image segmentation there we have to segment one image to another. And this is the foundation. Amazed!!

man, I love your videos, your voice and your vibe... you are amazing!!

17:28 , for a moment i thought he said : "If you wanna die" .. he got me excited

Respect Sir Isaac Newton

Great video! it's always nice to see real world applications for DEq. I believe that's when the "ah ha" moment happens. The chain with the barbell equation was one i used when i worked at the U of Tampa Human Performance Lab.

Thank you for this video. I opted out of calculus in college and always regretted it and this is the first video I’ve found that explains what differentia equations are used for without assuming the viewer already knows a ton of calculus-specific vocabulary. Thanks for helping me understand how they work a little better.

Differential Equations: *everything* Literally every student: *confused screaming*

2:40 OMG I love that show. I wish there were more math related shows

Wow! You actually made me excited about differential equations. Great job!!!

your efforts are easily visible through your eyes. great work

Up until Calc2, I had been a natural in math. But it stopped clicking sometime around when we were studying integrals, logarithms, and series. The nail in the coffin was the project I chose to try, which was to describe the curve of an archery bow as it was being drawn, relating the bow length, arc, and draw length or something like that. I flunked that project. I've since looked it up and found that I had bitten off more than I could chew because the related math and physics were a bit beyond what I had been studying at the time. But I didn't know that because my teacher didn't preview our project ideas to make sure we were on topic. So that's my advice to any would-be teachers reading. Be proactive with your students, ask them for updates on their semester projects and ask if they need any help. Don't just throw them in the deep end with no support.

My signals and systems course is like a kick in the face that says "hey man! Heres why you shouldve payed attention in DE!"

Amazing! Great job, loved it

Right on. Thanks for sharing.

Really nice video! Makes me miss teaching DEs... I wish we didn't focus so much on analytic solutions, but elementary is elementary. I wish there was more money in physics based solvers, I would love to find a job where I can build physics-based models like I did back in school...

I got a B+ in differential equations this semester. It was super easy, I really enjoyed it!

Amazingly explained, congrats!

Thank you so much, really great video, knowing why you do something is a must to learn something, generations of students have grown up not knowing why they learned something so gave it little effort,

what is the dot product of velocity and position vectors?

I tried solving my own version of your equation at 2:15, but with the area = the square of the arclength. It got very messy, ended up with a very nonlinear second order diff eq that looks hard to even numerically solve. I wish more diff eq's were easy to solve analytically.

Best video i watched this year। Thankyou for such videos, they inspires us towards science since we come to know why we learn all these and where we will be implementing them in future.

OMG The pursuit curves were freaking awesome!

Universe: BOOM! Here's a bunch of stuff. Us: Cool, how's it all work? Universe: BOOM! Here's Differential equations. Me: Cool, solve mine too.

I dont mind math, however I just dont like it when I make a tiny mistake that shoots my answer to a tangent

awesome! I like the applications, gives me a sense of all that what's happening with diff. eqs.

Interesting presentation. Great explaination. Helps to clear the mind

Finally i got answers which i can't get from schools and colleges

1:25 I lost it here

inb4 'If only schools taught like this', 'I wish they'd just play this videos in school' etc. - You can't learn to any significant degree from youtube videos. You like them because they are entertainment. Not everything can be made interesting. Good video.

I wish I had RU-vid when I was studying aeronautical engineering in the late 1988 to 1992. Compared to my university lecturers, Zach Star makes a differential equation look very simple

Hi zach !Lots of engineering topic are taught without giving any inituation /application. .. I believe step by step you will cover whole engineering course and would be able to create new engineering course 😅 best of Luck. ..greetings from India

This far too detailed and advanced for my secondary school brain

This is what I needed in my whole 12th for intuition . Thanks

This video brought back old fond memories to me... I was good at mathematics, really good. But took a different path. Still miss it.

At 1:46 meaning of the differetial equation was unexpected! ! Thanks for that!!

Yeah this is when I tapped out of maths at university and did a double major in geology.

Nicely explained! Thanks

i saw this at the end of my requirements list after calc 3 and wondered what could be more difficult than calc 3... thank you for scaring me.