The Geometric Meaning of Differential Equations // Slope Fields, Integral Curves & Isoclines

[@acreativeprogrammer]

I still remember that one of my high school math teacher had said that "You don't need to understand how math works in order to use math, just like how you don't need to understand how a computer work in order to operate a computer." In my first year of university, I struggled to understand math, calculus specifically, for the first time. I chose to listen to what my math teacher had said, but that only made me start to hate math. All I want to say is thank you for explaining math so thoroughly that I can start to like math again.

[@quant-prep2843]

this guy is a godsent, whenever i want the intuition i just type "whatever the topic Dr. Trefor Bazett"

[@kingwoomy8781]

I plan on taking Discrete Mathematics and Differential Equations next semester, so thank you in advance for any help I get from your videos.

[@ukrug7078]

Did You pass the exams? :D

[@handrez.]

Regalon del manguaco

[@CodeJeffo]

King of explaining ODEs even to total dummy. Wonderful. Thank you so much that from few videos I learned more than from whole semester. I learned the intuition behind ODEs.

[@lunal419]

This playlist of ODE is amazing! I haven't realized the relation between analytic and geometric representations of ODE. Thank you.

[@georgesadler7830]

Professor Bazett, thank you for a solid explanation of Slopes Fields, Integral Curves and Isoclines in Differential Equations. From watching this video and doing problems, increases my understanding of this topic.

[@aclif0316]

Man, I really wish that I had watched this at the beginning of my Diff EQ class instead of three days before finals.

[@amkhrjee]

This was such a brilliant video! Really loved the GeoGebra applet! Played with it for a while and it really made me click the concepts! Thanks a ton for making this awesome series!

[@BrittniHamilton]

Thank you so much for these videos and your breakdown of such a complex subject. You got me through Calc III and will be the reason I pass Diff. Eq.

[@DrTrefor]

I'm so happy to hear it - good luck in DiffyQs!

[@sijosthomas6314]

The geometrical meaning of diff eqns- Thankyou very much for explaining the big picture.

[@items365]

Sir this course is getting interesting day by day.

[@DrTrefor]

Glad to hear that!

[@sanjeet.kushwaha]

Thankyou sir from India🇮🇳 I am a high school student and was looking for geometrical interpretation of differential equations of first oder first degree and I found this insane explanation of the that. Now I have got it👍

[@colorx6030]

This is interesting. Coming into Differential Equations, I've always thought they were just some tool to solve problems analytically. Didn't know there were geometric interpretations of this as well. That's fascinating. Thanks for the video!

[@ahmdkalef2252]

Excellent geometrical representation of important concept. Well done

[@DrTrefor]

Glad you liked it!

[@AnjaliSharma06]

thank you, professor, your videos really help build mathematical intuition :)

[@DrTrefor]

Glad to hear that!

[@pinklady7184]

Anjali Sharma I think so too.

[@chg691]

I'd been waiting for this concept and yaa finally I got ...Thank you sir

[@ozguraslan36]

Many thanks Dr. Bazett! You have explained a very complicated topic clearly and understandable .

[@sarvasvkakkar2545]

Sir, the video really helped me to understand the topic in possibly the best way ever! Sir do you teach mathematics in any university?

[@DrTrefor]

Thank you! Yes I'm a math professor at the university of victoria

[@MrZWolfy]

Dr. Bazett is back at it. Thank you Sir!

[@DrTrefor]

haha thank you!!

[@MrZWolfy]

@@DrTrefor Anytime! Every bit is a great one. The fact that plenty of them help with field theory analysis makes them even better. For visualising these things, I'll welcome all the help available. Wish you a good one!

[@saadhassan9469]

Beautiful lecture

[@maciekleszczynski8414]

Great simple explanation! thank you for the time you spent on doing this 🤗

[@joewatson8439]

Thank you! I'm having so much trouble remembering some of this stuff for my grad Math Methods (Physics) class...

[@DrTrefor]

that sounds like a fun class!

[@joewatson8439]

@@DrTrefor It would be if our pace wasn't hyperspeed. Our first exam is tomorrow and is over 9 chapters ranging from series expansion and basic linear algebra to ODE and everything in between...we have 50 min

[@zennologyofeverything7265]

@@joewatson8439 did you improve compared to day 1?

[@AbDullAHMoHAAmeD]

This is so good. Thanks a lot Dr. Trefor.

[@pipertripp]

Really enjoyed this. Slope fields are a neat concept. As part of the geometric meaning of ODEs will you be discussing phase spaces at all?

[@DrTrefor]

Thank you! Yup, but not for a bit until we upgrade to systems of ODEs

[@pipertripp]

@@DrTrefor Looking fwd to it.

[@playitback-os7mh]

explanatory and on point as always. but the algorithm should know that too :)

[@DrTrefor]

Haha oh the algorithm;)

[@bunkydunk7500]

Great video. I just started homework on this topic.

[@zennologyofeverything7265]

Good energy

[@telodemuestro7384]

Holy shit, you opened my mind with the slope field idea. You are great

[@j.o.5957]

Question to self: slope fields and vector fields... given an initial position, a certain point will follow a line in a vector field, being tangent to the derivative at each point. Meaning a vector field, in that sense, is like a slope field, just for a more complex function?

[@mubahaliqbal5063]

I really enjoyed this lecture

[@continnum_radhe-radhe]

Thank you very much sir 🔥🔥🔥

[@vedantkulkarni6971]

A small doubt. So the Slope field is basically the general solution of the ODE and when we consider a intial condition using the point x0 and y0 we get a particular solution which is the integral curve. So the Slope field is kind of a combination of all the possible integral curves that can be obtained from the ODE?

[@DrTrefor]

That’s a good way of thinking about it yes. You could start anywhere and the slope field tells you where to travel.

[@vedantkulkarni6971]

@@DrTrefor Okay! Got it. Really enjoying the course, thanks a lot!

[@econhelp583]

Great video! The visualizations provide valuable insight into the problem and solution.

[@zainahmed1994]

Amazingly put.

[@SphereofTime]

8:23 Some particular point in actual slope field

[@faisalal1189]

OMG!! Really thank you. It is so satisfying to know these info

[@SphereofTime]

1:336 direction field

[@transcendingvictor]

incredible video

[@physicsismyfiancee...1353]

Can you clear a concept about 'sign of angles ' and what is the effect of that in any mathematical relation ??? Plssss, 💝 from india . I will wait for your reply, plssss help me .

[@carultch]

Do you mean sign as in positive vs negative? That usually refers to which direction the angle is, relative to which direction we define as positive, and where it starts. It's a common convention that CCW angles are positive and CW angles are negative Or do you mean sine as in the function sine?

[@salvatoregiordano6816]

Beautiful explanation!

[@akif7699]

Sir, I have a question. Our lecturer said the initial value of the differential equation can't affect stability of critical points. But gpt says it can affect and if I didn't understand it wrong from the video, it can affect. Can you explain it?

[@Woodsford123]

I understand this!!!! Thanks!

[@chiko4536]

Are you going to do a series on PDE's after this one?

[@DrTrefor]

That's definitely the plan!

[@shivangsingh5834]

Hmm so this is how differential equations helps us to generalize the equation through slope field you change any initial conditions you get a associated curves wonderful

[@hasanainalkifaee619]

Thank you so much

[@mnada72]

So the integral curve is the solution that satisfies the initial conditions ?

[@spotlight1415]

a question - in differential equation, which one would be dependent variable? would it be the least order of y or some x variable. and if its x then in above example is it dx somehow?

[@fernandojubany687]

Thanks for your great videos on differential equations. Can you solve , using Laplace transform, differential equations not having initial conditions?, to get a function of x, with an added constant C at the end? Could you always solve ODE by using Laplace? Thanks I am a retired engineer, just having fun reviewing what I was taught in Chile, at Santa Maria University in the 70's. Now with color graphs, and moving equations!, so much fun. Thanks

[@carultch]

Q1: Yes, you can solve differential equations with the Laplace transform, even if you didn't have an initial condition. What you'd do, is assign a variable as a placeholder for the initial condition, when you first set it up, and proceed as if you knew the initial condition. At the end of the day, you'll have constants in terms of your initial conditions on the parts of the solution that are solutions to the homogeneous equation, and constants that are independent of the initial conditions for what remains. As an example, consider y"(t) + 5*y'(t) + 6*y(t) = 6 - 6*e^(-t). Let u = y(0), and let v = y'(0). When you solve this using the Laplace transform, you'll end up with: Y(s) = 1/s - 3/(s + 1) + (3*u + v + 3)/(s + 2) - (2*u + v + 1)/(s + 3) You'll see that two of the coefficients are fixed, and are Laplace transforms that relate to the function 6 - 6*e^(-t). This is the particular solution. You'll also see two coefficients that are functions of u & v, which are the two coefficients that are undetermined when initial conditions are not specified. This is the complementary (or homogeneous) part of the solution. So if we don't care about directly relating these to the initial conditions, we can simply replace these coefficients with arbitrary letters. This gives us a general solution of: y(t) = 1 - 3*e^(-t) + A*e^(-2*t) + B*e^(-3*t) Q2: No, you cannot always use the Laplace transform to solve an ODE. It works well for cases when you're working with functions that play nicely with the integral that defines the Laplace transform, like exponentials, sines, cosines, coshes, sinches, and polynomials, as well as impulse and step functions based on the above, but it has trouble with any other kind of function. I've searched for an example where the Laplace of natural log, or of a non-polynomial algebraic function would help you, and I have yet to find any. Some functions simply have no closed-form Laplace transform in elementary functions, even though they should, like the laplace of arctan(t). This requires introducing the Si(s) and Ci(s) functions to find it. You usually need to use variation of parameters to work with exotic functions like this. Secant and tangent make some of the easiest examples to demonstrate variation of parameters.

[@theunique140]

Thank you ☺️

[@cheesypizzajokes]

Thank you!

[@axbowf4924]

Well the beard is growing very well. Along with my concepts on mathematics

[@DrTrefor]

haha my wife says I REALLY need to trim it:D

[@pinklady7184]

Dr. Trefor Bazett How many pencils can your beard hold?

[@neelkanthpatel7570]

Sir can we get it for second derivative??

[@ΚωνσταντίνοςΛαζαρίδης-ξ9ι]

thanks!

[@sergiolucas38]

Great video :)

[@sijosthomas6314]

Could you please suggest any book that treat diff eqns in this perspective?

[@dannychenski687]

Thank you sir! Honestly your voice isn't my favorite and I got a bad impression from a video of yours I watched a while ago, but you were of great help this time! Bias is real, so I don't think I'm a bad example of the way some people think

[@joypalit6408]

thank you sir! how can i do this in case o 2nd order equation

[@mcaramesalbite]

could you tell me, please, what software are you using?

[@DrTrefor]

This is geogebra, it is great!

[@nicolabellemo3054]

What about a second order differential equation?

[@sukranochani5764]

Thankyou

[@DrTrefor]

You’re welcome 😊

[@sin3divcx]

So, what's the connection beetween the graph of f(x,y)?

[@DrTrefor]

Well if you wre to graph z=f(x,y) this would be three dimensional and basically the height would give the slope. But this isn't particularly helpful way of visualzing here.

[@Lionel_9]

Which CAS did u use ???

[@KINGSLPK]

Great!!!!!

[@jakubpacua2351]

Wow I'm 12 years old and I am learning differential equations

[@abdelrahmanabusiam2792]

What about second order differential equations? Or in general higher order DE's

[@DrTrefor]

One approach for second order is to plot phase portraits where you ignore the time depndecne, but look at y and y' on the two axises and plot that relationship.

[@khalilmohammed2297]

i pluged in one point i havent got the another point for the point on the curve

[@physicsismyfiancee...1353]

is there any difference between f(x,y) and f(y,x)?

[@DrTrefor]

nope, just convention to write it f(x,y)

[@faisaladams3282]

Sir I watched the videos and understood them but then I cant seem to solve question.. I am getting frustrated😥😪

[@سلمىالترهوني-ي1ق]

الترجمة الى العربية

[@tanmaydeshmukh3517]

First comment

[@jaw0449]

nope

[@DrTrefor]

nice!!

[@dsm5d723]

All me, YOU all have been warned. What about the million dollar rabbit lies to dig for? All me again. $$$$, if you want that metric.