Information about Ordinary Differential Equations - Part 1

1 00:00:00,457 –> 00:00:06,276 Hello and welcome to the new video series about ordinary differential equations.

2 00:00:07,114 –> 00:00:14,859 This will be a video series for everyone who is interested in the theory of differential equations and how to solve them.

3 00:00:15,059 –> 00:00:17,729 So essentially it’s a mathematics course,

4 00:00:17,757 –> 00:00:23,161 but it’s also important for a lot of other science subjects, where differential equations occur.

5 00:00:23,657 –> 00:00:31,101 Therefore you might have already seen a differential equation. For example f’=f is a common one.

6 00:00:31,301 –> 00:00:38,697 Roughly speaking this equation means that we search for a function f, that is differentiable and fulfills this equation.

7 00:00:38,897 –> 00:00:45,098 More precisely here it means that the derivative of the function f is equal to the function itself

8 00:00:45,298 –> 00:00:50,414 and then you might already find, that the exponential function does the job.

9 00:00:50,671 –> 00:00:55,386 So f(x) = e^x solves this differential equation.

10 00:00:55,929 –> 00:01:00,733 Ok, at this point I would say a lot of questions immediately occur.

11 00:01:01,300 –> 00:01:06,651 First, how can we solve such an equation if we don’t have any idea, what the solution could be?

12 00:01:06,971 –> 00:01:12,707 Second, in the case that find a solution, what can we say about the uniqueness of this solution?

13 00:01:13,200 –> 00:01:19,571 Moreover, besides the existence and uniqueness of solutions, you could also ask smart questions, like

14 00:01:19,586 –> 00:01:22,220 what is the domain of definition for a solution here

15 00:01:22,257 –> 00:01:24,597 and can it be different for different solutions?

16 00:01:24,900 –> 00:01:30,385 Indeed all these questions and many more, we will answer in this video course here

17 00:01:30,743 –> 00:01:37,975 and a good starting point would be to talk about the prerequisite you need to understand all the videos here.

18 00:01:38,175 –> 00:01:45,010 However before we do that, I really want to thank all the nice people who support me on Steady, via Paypal or by other means.

19 00:01:45,514 –> 00:01:51,976 This freely available video course only exists, because nice people support the production of it

20 00:01:52,176 –> 00:01:58,323 and indeed as a supporter you get rewards like quizzes and PDF versions for all the videos.

21 00:01:58,857 –> 00:02:01,963 To find them, just click the link in the description.

22 00:02:02,514 –> 00:02:09,138 Ok, now let’s talk about the requirements. The knowledge you need to understand ordinary differential equations.

23 00:02:09,800 –> 00:02:15,086 In fact there are 2 main ingredients here. On the one side you need to know what derivatives are.

24 00:02:15,214 –> 00:02:17,087 So you need the real analysis part

25 00:02:17,287 –> 00:02:21,549 and then you need to calculate with vectors. So you need the linear algebra part.

26 00:02:22,143 –> 00:02:26,942 For both topics I have video courses, where you can watch the relevant parts.

27 00:02:27,471 –> 00:02:35,090 However, if you are very new to mathematics, I would like to suggest that you first watch my “start learning mathematics” course.

28 00:02:35,686 –> 00:02:43,769 There I tell you about typical notations, what numbers are, what sets are. So all the foundations you need to understand the rest here.

29 00:02:44,371 –> 00:02:49,748 Ok, so now you know what we need and you see, we don’t need a lot of multivariable calculus.

30 00:02:49,948 –> 00:02:53,384 Simply because we will not talk about partial differential equations.

31 00:02:53,886 –> 00:02:59,559 So we will not consider partial derivatives in the equations just ordinary derivatives.

32 00:03:00,271 –> 00:03:04,594 Indeed this makes the whole topic much easier to deal with.

33 00:03:05,043 –> 00:03:12,029 However in order to motivate you why ordinary differential equations are still interesting and very important

34 00:03:12,114 –> 00:03:14,457 let’s look at some other examples.

35 00:03:15,057 –> 00:03:21,886 So for example you might know that in physics one uses dots for denoting the time derivative.

36 00:03:22,571 –> 00:03:27,071 Hence this here is the second derivative of the function x(t)

37 00:03:27,314 –> 00:03:30,928 and now this should be equal to omega squared x.

38 00:03:31,714 –> 00:03:37,529 Indeed, in physics this equation here describes a so called harmonic oscillator

39 00:03:37,829 –> 00:03:43,343 and depending on how much you want to calculate with complex numbers, we have a minus sign here or not

40 00:03:43,543 –> 00:03:49,972 and there I should tell you, if you don’t know complex numbers, you should check out my “start learning mathematics” series.

41 00:03:50,671 –> 00:03:55,693 However in the end here complex numbers will only be important for some calculations.

42 00:03:56,157 –> 00:04:01,036 Ok, so now we can consider another example, also given by physics.

43 00:04:01,643 –> 00:04:07,881 Indeed, this one is easy to explain. You just have 3 planets here and they will gravity.

44 00:04:08,357 –> 00:04:11,412 So simply said they will attract each other

45 00:04:11,843 –> 00:04:20,600 and now if you describe the one planet in the 3 dimensions with a vector x and the one with a vector y and the 3rd one with a vector z,

46 00:04:20,671 –> 00:04:23,682 then we can use newton’s gravity.

47 00:04:24,057 –> 00:04:30,191 So the mass m times x, second derivative is equal to the gravity force.

48 00:04:30,757 –> 00:04:36,393 However, this force F now is a function that depends on all the positions here.

49 00:04:36,657 –> 00:04:42,086 Hence the equations for the different planets here are connected by this force.

50 00:04:42,571 –> 00:04:48,381 So you see, we have 9 differential equations that are completely cross linked here.

51 00:04:49,086 –> 00:04:53,168 Hence we would say, we have a whole system of differential equations

52 00:04:53,368 –> 00:04:55,571 and we want to solve the whole system.

53 00:04:55,971 –> 00:05:01,919 Moreover, you see here in contrast to the first example, we have second order derivatives here.

54 00:05:02,543 –> 00:05:10,266 However, we will see that for the theory of ordinary differential equations the order of the derivatives is not so important.

55 00:05:10,914 –> 00:05:16,033 Ok and this brings me to a short overview what we will do in this course.

56 00:05:16,233 –> 00:05:21,821 So the first I already told you. We will talk about systems of differential equations.

57 00:05:22,021 –> 00:05:29,189 Moreover in the future we will abbreviate ordinary differential equations by ODE.

58 00:05:29,529 –> 00:05:35,514 Ok, then after this abstract theory we will talk about some practical solution methods.

59 00:05:36,100 –> 00:05:41,542 So we will look at the procedure, how to solve such a system of ODE

60 00:05:42,000 –> 00:05:48,784 and then we will go back to the theory again and prove the existence and uniqueness of solutions.

61 00:05:48,984 –> 00:05:54,631 In particular we will see, what are the correct assumptions we need to have uniqueness

62 00:05:54,831 –> 00:06:00,076 and then in the end we will also talk about linear ODE.

63 00:06:00,529 –> 00:06:05,312 There the so called matrix exponential function will play a crucial role.

64 00:06:05,971 –> 00:06:12,342 So this is the plan for this course and I would say, let’s start with the important definitions in the next video.

65 00:06:12,800 –> 00:06:16,236 Therefore let’s meet there and have a nice day. Bye!

Q1: What is not a correct definition of $f^\prime(x_0)$ for a function $f: \mathbb{R} \rightarrow \mathbb{R}$?

A1: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$.

A2: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x_0) - f(x)}{x_0 - x}$.

A3: $\displaystyle \lim_{z \rightarrow x_0} \frac{f(x_0) - f(z)}{x_0 - z}$.

A4: $\displaystyle \lim_{x_0 \rightarrow x} \frac{f(x_0) - f(x)}{x - x_0}$.

A5: $\displaystyle \lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_0)}{x_n - x_0}$ if we get the same value for each sequence $x_n \xrightarrow{n \rightarrow \infty} x_0$.

Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = |x|$ and $x_0 = 1$. What is $f^\prime(x_0)$?

A1: 0

A2: -1

A3: 1

A4: It does not exist.

Q3: Which of the following functions does not satisfy the differential equation $f^\prime = f$?

A1: $ \exp(x) $

A2: $ 3 \exp(x) $

A3: $ 0 $

A4: $ \exp(x) +1 $