The Simplest Ordinary Differential Equation (ODE) and Its Exponential Solution

Here we introduce the simplest linear, first-order ordinary differential equation, dx/dt = constant * x, using intuitive examples like the growth of a bunny population, radioactive decay, and compound interest in the bank. We solve this equation and find that the solutions are given by exponential functions in time. This is the basis for all of the differential equations in this series.
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This video was produced at the University of Washington
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0:58 Example: Bunny Population Growth
6:33 Solving this Differential Equation
15:03 What is Euler's Number 'e'? Example: Compound Interest
27:25 Loan Interest as a Differential Equation
31:23 Example: Radioactive Decay
36:32 Example: Thermal Runaway in Electronics

[@kumbajobe6864]

I have been attending lectures for more than a month and I had no idea what the lecturer was saying until I came across your videos. You are great 👍 sir.

[@Eigensteve]

Glad they are helpful!

[@fatihsarikoc570]

Thank you Steve from Türkiye, for your effort and lovely introduction to DEs.

[@mberoakoko24]

Your tutorials are invaluable and I'd like to thank you for making complex subjects available for everyone. Thank you sir 😊

[@elyezerf58]

The content is amazing and it is impressive the fact that the professor needs to write everything in contrary direction to provide amazing views

[@Eigensteve]

Thanks!

[@AI_BotBuilder]

You just connected some missing math piece in my life, I’ve been trying to recall the day when I was introduced to e in school, turns out I was just taught how to solve it but it never made sense to me until today. Thankyou so much❤❤❤❤

[@heyman620]

I think it's the best course I have seen for any math subject - and I studied CS (grad) and some grad math courses. But I have a comment. I felt extremely uncomfortable when I saw you multiplying by dt (what can I do, studied many courses with the Math department) - so it's important to note that It does not work, in general, just in some cases. It's important since if it seems weird to me (and probably other people) we have to look it up online, and answers are of very low quality (people that are not skilled enough to answer it rigorously - I would rather hear that information from you, that's why I don't do it here as well). I think it's due to the fact that I studied calc with Math department and we were not allowed to use tricks we can't prove (Differential forms is more "niche"). Thanks you so much!

[@lernenmitrobin]

dear Steve, you make a very good job and have a clear explanation in every single lecture on your channel. Don't change anything and keep that direction. Regards from Germany - Robin

[@rushabhyeshwante]

Thank you for this lecture. I understood, what exactly is exponential 'e' term. In my high school, we used to just the formula of doing it's integration and differentiation. The examples of compound interest, temperature runaway was good and I could relate it with my day-to day life.

[@rocketman99]

Steve is a legend

[@karamany9870]

Prof. you really love Euler :D

[@mohamadismailbasha6083]

Wonderful lectures. Thank you very much for sharing

[@nHans]

And that's exactly why I've always advocated teaching financial literacy to students-starting in middle school itself! So many graduates are entering life without knowing the basics of financial management. Not understanding how credit cards work-aka compound interest-has literally bankrupted so many people!

[@starriet]

note for thinking) _if_ the annual interest rate is exactly 1+r, not e^r, then x(t) would be (1+r)^t times x(0), where the unit of t is a year. Then, dx/dt would be ln(1+r) times x, not r times x. This is also related to 29:10~, where it's not continuous but rather discrete(delta x and delta t are not infinitesimally small). Prof.Steve mentions this too 29:28~.

[@perrylau6334]

very clear and very interesting examples!!!!!

[@SO-th8gc]

Excellent explanation, really enjoying it.

[@edwardgongsky8540]

I finally understand what 'e' is! thanks professor!

[@nhlhockeyallstar]

thank you professor

[@Jaylooker]

Excellent explanation

[@user-rn7tq6bn7f]

very wonderful , thank you very much

[@euglossine4ever]

Thank you Dr Steve ❤

[@AlessandroZir]

thanks! there is some more "conceptual" explanation of "e" I heard somewhere, which I think is also useful to get an intuition of what "e" is; I don't know if I can express it myself, but perhaps goes like this: "e" has to do with some kind of proportionality, with how anything can grow or degrade with respect of what that thing already is; things don't generally grow or degrade entirely out of proportion, even when they grow or degrade a lot; ;

[@MrFazeFaze]

Thank you for doing this series. Would you be interested in doing a series on numerical methods for PDEs like FVM?

[@Eltrissi]

Just another great lec... 👍👍❤️

[@curtpiazza1688]

Great lecture! Cool examples! 😊

[@shilpajain2329]

too good.! enjoyed the lecture and they way the things are explained. My question is how do u write in opposite direction so easily.... :)

[@rafaelstv]

I believe it would be best to explicitly write dx/dt = λx(t) instead ot dx/dt = λx for the derivative. It is confusing since λ is a constant, and when we set λ=2, we obtain 2x. And that is the derivative of a polynomial, x^2, and not an exponencial.

[@user-js6yn2hc5p]

Thanks 🙏 By the way you have a sweet sense of humor professor :)

[@pappaflammyboi5799]

A more technical view, or shall we say, more accurate description of radioactive decay, is not the loss of mass of the starting material, albeit there is some of that happening if the radioactive particles (i.e. alphas, betas(+-), gammas, x-rays, neutrons, protons, and neutrinos) emitting away from the isotope aren't constrained to your environmental frame of reference. But, what is actually happening is the starter material is changing thru transmutation from one isotope to another and emitting particles in the process. The majority of the mass still exists (except for the unconstrained emitting particles), it's just a different element now after the decay. Just a small nitpick.

[@lioneloddo]

Ah Ok !!! This is that the profond meaning of the exponential !!!! The slope is equal (or proportional) to the position !! The more the position invreases, the more the slope increases ! The differential equation says just that !!! The slope is equal or proportional to the position !!!!

[@channingparker9431]

Is there a differential equation to figure out the percentage of the population that may be watching and stalking a particular individual at any given moment in space time?

[@deannawright2445]

Can I draw your attention to vol 37 British Journal for the History of Mathematics, Why was Leonhard Euler blind it may have actually been brucellosis...thanks for the maths videos really enjoying them.

[@-mwolf]

you can also import math; math.exp instead of numpy

[@yogeshriyat]

What’s the best way to review this method one integration by parts? I do have the Thomas book.

[@shashidharreddy2959]

please make some lectures on tensors and coding the tensors.

[@oraz.]

Why are there multiple channels with different people writing similarly on a black background about dynamical systems? Just curious.

[@samg9908]

Shouts out to the algorithm

[@bradygenz5388]

What is the lambda constant for Killer Bunnies of Caer Bannog?

[@KevinJDenny]

In discrete time I’d write it as x(t)-x(t-1)= r.x(t-1) , an AR(1) process

[@jawadmansoor6064]

7:30 how can you divide both sides with 'x' and then multiply with 'dt'? And what is even the equation at 7:43?

[@gustavgille9323]

They are both valid mathematical operations, by dividing both sides (and later multiplying dt), he only rearranges the equality. This is done as to separate the variables so they can be integrated individually, with the integration applied on both sides of the equality.

[@chipfoo5115]

Whose...join this classes....will think twice about taking loan..

[@fabiofarina9579]

Malthus likes this item. Now Vito Volterra can't wait to take the stage

[@leolima75]

This problem's solution is only valid for x >= 2 at t = 0 ... basically because we can't reverse time (yet) and, even if we could, we wouldn't get fractionary bunnys and, obviously, we need at least 2 bunnys to generate more bunnys. Which also reminds me that the solution to the bunnys ODE is also only defined in x belonging to the group of positive natural numbers.

[@lioneloddo]

2**x = exp(x.ln(2)) So ln(2) can be seen as a simple coefficient lambda.

[@leolima75]

@@lioneloddo and what does that have to with anything I posted ? I'll translate my post to you: oftentimes mathematical formulations are meaningless without proper physical interpretation.

[@nHans]

​@@leolima75 Do watch the video till the end. Around 29:00, he addresses the exact points that you're making: When dealing with a discrete system (i.e., no fractions allowed), when and why is it okay to use continuously-varying quantities to model its behavior. He also makes a great point about carefully examining our assumptions. You do understand that everything in our physical world is discrete, right? Nothing is continuous. Space, time, matter, energy-everything is quantized. Prof. Brunton himself mentions Planck Time when discussing compound interest. Yet, we can successfully model it using continuous variables. Again-watch the whole video till the end.

[@leolima75]

@@nHans did I contradict him in any way ?

[@dhnguyen68]

I also learnt dx/ dt is equal x prime (x’)

[@LucyWilliams-lu7ow]

My favorite part is when he erases fast

[@carstenmeyer7786]

7:22 There is another option -- just as simple, but actually rigorous. Instead of "splitting *dx/dt* ", just divide by *"x ≄ 0"* to get: *𝜆 = (dx/dt) / x = d/dt ln|x(t)| // t -> t', ∫ ... dt'* Use the fundamental theorem of calculus (FTC) on the RHS to get *𝜆t + C = ln|x(t)| => x(t) = e^{𝜆t} * x0 // x0 = ∓e^{C}*

[@TheMailbawx]

import numpy as np import matplotlib.pyplot as plt r = 1.5 N = 100 for n in range(1,N): i = np.arange(n+1) array = (1+r/n)**i plt.plot(i/n, array, '-') x = np.linspace(0,1) plt.plot(x, np.exp(r*x), 'k--') plt.show()