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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.96,0:00:02.98,Default,,0000,0000,0000,,I'll now introduce you\Nto the concept
Dialogue: 0,0:00:02.98,0:00:05.46,Default,,0000,0000,0000,,of the Laplace Transform.
Dialogue: 0,0:00:05.46,0:00:09.93,Default,,0000,0000,0000,,And this is truly one of the\Nmost useful concepts that
Dialogue: 0,0:00:09.93,0:00:13.84,Default,,0000,0000,0000,,you'll learn, not just in\Ndifferential equations, but
Dialogue: 0,0:00:13.84,0:00:15.10,Default,,0000,0000,0000,,really in mathematics.
Dialogue: 0,0:00:15.10,0:00:18.05,Default,,0000,0000,0000,,And especially if you're going\Nto go into engineering, you'll
Dialogue: 0,0:00:18.05,0:00:20.61,Default,,0000,0000,0000,,find that the Laplace Transform,\Nbesides helping you
Dialogue: 0,0:00:20.61,0:00:25.48,Default,,0000,0000,0000,,solve differential equations,\Nalso helps you transform
Dialogue: 0,0:00:25.48,0:00:30.21,Default,,0000,0000,0000,,functions or waveforms from\Nthe time domain to the
Dialogue: 0,0:00:30.21,0:00:33.17,Default,,0000,0000,0000,,frequency domain, and study\Nand understand a
Dialogue: 0,0:00:33.17,0:00:34.74,Default,,0000,0000,0000,,whole set of phenomena.
Dialogue: 0,0:00:34.74,0:00:36.48,Default,,0000,0000,0000,,But I won't get into\Nall of that yet.
Dialogue: 0,0:00:36.48,0:00:38.92,Default,,0000,0000,0000,,Now I'll just teach\Nyou what it is.
Dialogue: 0,0:00:38.92,0:00:40.17,Default,,0000,0000,0000,,Laplace Transform.
Dialogue: 0,0:00:42.96,0:00:45.10,Default,,0000,0000,0000,,I'll teach you what it is, make\Nyou comfortable with the
Dialogue: 0,0:00:45.10,0:00:48.33,Default,,0000,0000,0000,,mathematics of it and then in\Na couple of videos from now,
Dialogue: 0,0:00:48.33,0:00:52.22,Default,,0000,0000,0000,,I'll actually show you how it\Nis useful to use it to solve
Dialogue: 0,0:00:52.22,0:00:53.18,Default,,0000,0000,0000,,differential equations.
Dialogue: 0,0:00:53.18,0:00:55.20,Default,,0000,0000,0000,,We'll actually solve some of the\Ndifferential equations we
Dialogue: 0,0:00:55.20,0:00:56.80,Default,,0000,0000,0000,,did before, using the\Nprevious methods.
Dialogue: 0,0:00:56.80,0:00:59.47,Default,,0000,0000,0000,,But we'll keep doing it, and\Nwe'll solve more and more
Dialogue: 0,0:00:59.47,0:01:01.00,Default,,0000,0000,0000,,difficult problems.
Dialogue: 0,0:01:01.00,0:01:02.89,Default,,0000,0000,0000,,So what is the Laplace\NTransform?
Dialogue: 0,0:01:02.89,0:01:08.58,Default,,0000,0000,0000,,Well, the Laplace Transform,\Nthe notation is the L like
Dialogue: 0,0:01:08.58,0:01:12.06,Default,,0000,0000,0000,,Laverne from Laverne\Nand Shirley.
Dialogue: 0,0:01:12.06,0:01:15.04,Default,,0000,0000,0000,,That might be before many\Nof your times, but
Dialogue: 0,0:01:15.04,0:01:16.86,Default,,0000,0000,0000,,I grew up on that.
Dialogue: 0,0:01:16.86,0:01:20.69,Default,,0000,0000,0000,,Actually, I think it was even\Nreruns when I was a kid.
Dialogue: 0,0:01:20.69,0:01:22.77,Default,,0000,0000,0000,,So Laplace Transform\Nof some function.
Dialogue: 0,0:01:22.77,0:01:25.17,Default,,0000,0000,0000,,And here, the convention,\Ninstead of saying f of x,
Dialogue: 0,0:01:25.17,0:01:26.59,Default,,0000,0000,0000,,people say f of t.
Dialogue: 0,0:01:26.59,0:01:30.12,Default,,0000,0000,0000,,And the reason is because in\Na lot of the differential
Dialogue: 0,0:01:30.12,0:01:32.19,Default,,0000,0000,0000,,equations or a lot of\Nengineering you actually are
Dialogue: 0,0:01:32.19,0:01:34.36,Default,,0000,0000,0000,,converting from a function\Nof time to
Dialogue: 0,0:01:34.36,0:01:35.63,Default,,0000,0000,0000,,a function of frequency.
Dialogue: 0,0:01:35.63,0:01:37.30,Default,,0000,0000,0000,,And don't worry about\Nthat right now.
Dialogue: 0,0:01:37.30,0:01:40.18,Default,,0000,0000,0000,,If it confuses you.
Dialogue: 0,0:01:40.18,0:01:43.02,Default,,0000,0000,0000,,But the Laplace Transform\Nof a function of t.
Dialogue: 0,0:01:43.02,0:01:47.82,Default,,0000,0000,0000,,It transforms that function into\Nsome other function of s.
Dialogue: 0,0:01:47.82,0:01:49.41,Default,,0000,0000,0000,,and And does it do that?
Dialogue: 0,0:01:49.41,0:01:53.15,Default,,0000,0000,0000,,Well actually, let me just do\Nsome mathematical notation
Dialogue: 0,0:01:53.15,0:01:56.46,Default,,0000,0000,0000,,that probably won't\Nmean much to you.
Dialogue: 0,0:01:56.46,0:01:57.80,Default,,0000,0000,0000,,So what does it transform?
Dialogue: 0,0:01:57.80,0:01:59.63,Default,,0000,0000,0000,,Well, the way I think of\Nit is it's kind of
Dialogue: 0,0:01:59.63,0:02:00.87,Default,,0000,0000,0000,,a function of functions.
Dialogue: 0,0:02:00.87,0:02:05.22,Default,,0000,0000,0000,,A function will take you from\None set of-- well, in what
Dialogue: 0,0:02:05.22,0:02:08.01,Default,,0000,0000,0000,,we've been dealing with-- one\Nset of numbers to another set
Dialogue: 0,0:02:08.01,0:02:09.06,Default,,0000,0000,0000,,of numbers.
Dialogue: 0,0:02:09.06,0:02:11.99,Default,,0000,0000,0000,,A transform will take you from\None set of functions to
Dialogue: 0,0:02:11.99,0:02:13.10,Default,,0000,0000,0000,,another set of functions.
Dialogue: 0,0:02:13.10,0:02:14.14,Default,,0000,0000,0000,,So let me just define this.
Dialogue: 0,0:02:14.14,0:02:23.31,Default,,0000,0000,0000,,The Laplace Transform for our\Npurposes is defined as the
Dialogue: 0,0:02:23.31,0:02:24.51,Default,,0000,0000,0000,,improper integral.
Dialogue: 0,0:02:24.51,0:02:27.79,Default,,0000,0000,0000,,I know I haven't actually done\Nimproper integrals just yet,
Dialogue: 0,0:02:27.79,0:02:29.89,Default,,0000,0000,0000,,but I'll explain them\Nin a few seconds.
Dialogue: 0,0:02:29.89,0:02:36.43,Default,,0000,0000,0000,,The improper integral from 0 to\Ninfinity of e to the minus
Dialogue: 0,0:02:36.43,0:02:43.60,Default,,0000,0000,0000,,st times f of t-- so whatever's\Nbetween the Laplace
Dialogue: 0,0:02:43.60,0:02:49.17,Default,,0000,0000,0000,,Transform brackets-- dt.
Dialogue: 0,0:02:49.17,0:02:51.19,Default,,0000,0000,0000,,Now that might seem very\Ndaunting to you and very
Dialogue: 0,0:02:51.19,0:02:54.12,Default,,0000,0000,0000,,confusing, but I'll now do\Na couple of examples.
Dialogue: 0,0:02:54.12,0:02:55.66,Default,,0000,0000,0000,,So what is the Laplace\NTransform?
Dialogue: 0,0:02:55.66,0:02:57.95,Default,,0000,0000,0000,,Well let's say that f\Nof t is equal to 1.
Dialogue: 0,0:02:57.95,0:03:00.30,Default,,0000,0000,0000,,So what is the Laplace\NTransform of 1?
Dialogue: 0,0:03:04.18,0:03:07.66,Default,,0000,0000,0000,,So if f of t is equal to 1--\Nit's just a constant function
Dialogue: 0,0:03:07.66,0:03:14.28,Default,,0000,0000,0000,,of time-- well actually, let me\Njust substitute exactly the
Dialogue: 0,0:03:14.28,0:03:15.13,Default,,0000,0000,0000,,way I wrote it here.
Dialogue: 0,0:03:15.13,0:03:18.91,Default,,0000,0000,0000,,So that's the improper integral\Nfrom 0 to infinity of
Dialogue: 0,0:03:18.91,0:03:24.64,Default,,0000,0000,0000,,e to the minus st\Ntimes 1 here.
Dialogue: 0,0:03:24.64,0:03:29.00,Default,,0000,0000,0000,,I don't have to rewrite it here,\Nbut there's a times 1dt.
Dialogue: 0,0:03:29.00,0:03:32.27,Default,,0000,0000,0000,,And I know that infinity is\Nprobably bugging you right
Dialogue: 0,0:03:32.27,0:03:34.48,Default,,0000,0000,0000,,now, but we'll deal\Nwith that shortly.
Dialogue: 0,0:03:34.48,0:03:35.62,Default,,0000,0000,0000,,Actually, let's deal with\Nthat right now.
Dialogue: 0,0:03:35.62,0:03:40.66,Default,,0000,0000,0000,,This is the same thing\Nas the limit.
Dialogue: 0,0:03:40.66,0:03:48.87,Default,,0000,0000,0000,,And let's say as A approaches\Ninfinity of the integral from
Dialogue: 0,0:03:48.87,0:03:57.40,Default,,0000,0000,0000,,0 to Ae to the minus st. dt.
Dialogue: 0,0:03:57.40,0:03:59.41,Default,,0000,0000,0000,,Just so you feel a little bit\Nmore comfortable with it, you
Dialogue: 0,0:03:59.41,0:04:01.64,Default,,0000,0000,0000,,might have guessed that this\Nis the same thing.
Dialogue: 0,0:04:01.64,0:04:04.56,Default,,0000,0000,0000,,Because obviously you can't\Nevaluate infinity, but you
Dialogue: 0,0:04:04.56,0:04:07.41,Default,,0000,0000,0000,,could take the limit as\Nsomething approaches infinity.
Dialogue: 0,0:04:07.41,0:04:09.88,Default,,0000,0000,0000,,So anyway, let's take the\Nanti-derivative and evaluate
Dialogue: 0,0:04:09.88,0:04:12.73,Default,,0000,0000,0000,,this improper definite\Nintegral, or
Dialogue: 0,0:04:12.73,0:04:13.81,Default,,0000,0000,0000,,this improper integral.
Dialogue: 0,0:04:13.81,0:04:17.30,Default,,0000,0000,0000,,So what's anti-derivative\Nof e to the minus st
Dialogue: 0,0:04:17.30,0:04:19.34,Default,,0000,0000,0000,,with respect to dt?
Dialogue: 0,0:04:19.34,0:04:28.62,Default,,0000,0000,0000,,Well it's equal to minus 1/s\Ne to the minus st, right?
Dialogue: 0,0:04:28.62,0:04:30.64,Default,,0000,0000,0000,,If you don't believe me, take\Nthe derivative of this.
Dialogue: 0,0:04:30.64,0:04:32.07,Default,,0000,0000,0000,,You'd take minus s times that.
Dialogue: 0,0:04:32.07,0:04:34.50,Default,,0000,0000,0000,,That would all cancel out, and\Nyou'd just be left with e to
Dialogue: 0,0:04:34.50,0:04:36.46,Default,,0000,0000,0000,,the minus st. Fair enough.
Dialogue: 0,0:04:39.72,0:04:42.41,Default,,0000,0000,0000,,Let me delete this here,\Nthis equal sign.
Dialogue: 0,0:04:42.41,0:04:45.89,Default,,0000,0000,0000,,Because I could actually use\Nsome of that real estate.
Dialogue: 0,0:04:45.89,0:04:51.43,Default,,0000,0000,0000,,We are going to take the limit\Nas A approaches infinity.
Dialogue: 0,0:04:51.43,0:04:53.33,Default,,0000,0000,0000,,You don't always have to do\Nthis, but this is the first
Dialogue: 0,0:04:53.33,0:04:54.65,Default,,0000,0000,0000,,time we're dealing with\Nimproper intergrals.
Dialogue: 0,0:04:54.65,0:04:57.27,Default,,0000,0000,0000,,So I figured I might as\Nwell remind you that
Dialogue: 0,0:04:57.27,0:04:59.34,Default,,0000,0000,0000,,we're taking a limit.
Dialogue: 0,0:04:59.34,0:05:01.03,Default,,0000,0000,0000,,Now we took the anti-derivative.
Dialogue: 0,0:05:01.03,0:05:04.96,Default,,0000,0000,0000,,Now we have to evaluate it at A\Nminus the anti-derivative
Dialogue: 0,0:05:04.96,0:05:06.05,Default,,0000,0000,0000,,evaluate it at 0,
Dialogue: 0,0:05:06.05,0:05:08.74,Default,,0000,0000,0000,,and then take the limit of\Nwhatever that ends up being as
Dialogue: 0,0:05:08.74,0:05:09.71,Default,,0000,0000,0000,,A approaches infinity.
Dialogue: 0,0:05:09.71,0:05:17.49,Default,,0000,0000,0000,,So this is equal to the limit\Nas A approaches infinity.
Dialogue: 0,0:05:17.49,0:05:17.75,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:05:17.75,0:05:24.55,Default,,0000,0000,0000,,If we substitute A in here\Nfirst, we get minus 1/s.
Dialogue: 0,0:05:24.55,0:05:26.96,Default,,0000,0000,0000,,Remember we're, dealing\Nwith t.
Dialogue: 0,0:05:26.96,0:05:30.22,Default,,0000,0000,0000,,We took the integral\Nwith respect to t.
Dialogue: 0,0:05:30.22,0:05:36.63,Default,,0000,0000,0000,,e to the minus sA, right?
Dialogue: 0,0:05:36.63,0:05:38.65,Default,,0000,0000,0000,,That's what happens when\NI put A in here.
Dialogue: 0,0:05:38.65,0:05:41.35,Default,,0000,0000,0000,,Minus -
Dialogue: 0,0:05:41.35,0:05:44.97,Default,,0000,0000,0000,,Now what happens when I put\Nt equals 0 in here?
Dialogue: 0,0:05:44.97,0:05:47.83,Default,,0000,0000,0000,,So when t equals 0, it becomes\Ne to the minus s times 0.
Dialogue: 0,0:05:47.83,0:05:49.32,Default,,0000,0000,0000,,This whole thing becomes 1.
Dialogue: 0,0:05:49.32,0:05:51.19,Default,,0000,0000,0000,,And I'm just left\Nwith minus 1/s.
Dialogue: 0,0:05:57.80,0:05:58.45,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:05:58.45,0:06:01.00,Default,,0000,0000,0000,,And then let me scroll\Ndown a little bit.
Dialogue: 0,0:06:01.00,0:06:02.49,Default,,0000,0000,0000,,I wrote a little bit bigger\Nthan I wanted
Dialogue: 0,0:06:02.49,0:06:03.77,Default,,0000,0000,0000,,to, but that's OK.
Dialogue: 0,0:06:03.77,0:06:10.16,Default,,0000,0000,0000,,So this is going to be the limit\Nas A approaches infinity
Dialogue: 0,0:06:10.16,0:06:20.64,Default,,0000,0000,0000,,of minus 1/s e to the\Nminus sA minus minus 1/s.
Dialogue: 0,0:06:20.64,0:06:24.78,Default,,0000,0000,0000,,So plus 1/s.
Dialogue: 0,0:06:24.78,0:06:26.17,Default,,0000,0000,0000,,So what's the limit as A\Napproaches infinity?
Dialogue: 0,0:06:26.17,0:06:28.15,Default,,0000,0000,0000,,Well what's this term\Ngoing to do?
Dialogue: 0,0:06:28.15,0:06:34.35,Default,,0000,0000,0000,,As A approaches infinity, if\Nwe assume that s is greater
Dialogue: 0,0:06:34.35,0:06:37.81,Default,,0000,0000,0000,,than 0-- and we'll make that\Nassumption for now.
Dialogue: 0,0:06:37.81,0:06:39.00,Default,,0000,0000,0000,,Actually, let me write\Nthat down explicitly.
Dialogue: 0,0:06:39.00,0:06:41.95,Default,,0000,0000,0000,,Let's assume that s\Nis greater than 0.
Dialogue: 0,0:06:41.95,0:06:45.32,Default,,0000,0000,0000,,So if we assume that s is\Ngreater than 0, then as A
Dialogue: 0,0:06:45.32,0:06:47.87,Default,,0000,0000,0000,,approaches infinity, what's\Ngoing to happen?
Dialogue: 0,0:06:47.87,0:06:53.21,Default,,0000,0000,0000,,Well this term is going to go to\N0, right? e to the minus--
Dialogue: 0,0:06:53.21,0:06:55.64,Default,,0000,0000,0000,,a googol is a very,\Nvery small number.
Dialogue: 0,0:06:55.64,0:07:00.52,Default,,0000,0000,0000,,And an e to the minus googolplex\Nis an even smaller number.
Dialogue: 0,0:07:00.52,0:07:04.53,Default,,0000,0000,0000,,So then this e to the minus\Ninfinity approaches 0, so this
Dialogue: 0,0:07:04.53,0:07:05.92,Default,,0000,0000,0000,,term approaches 0.
Dialogue: 0,0:07:05.92,0:07:08.85,Default,,0000,0000,0000,,This term isn't affected because\Nit has no A in it, so
Dialogue: 0,0:07:08.85,0:07:12.42,Default,,0000,0000,0000,,we're just left with 1/s.
Dialogue: 0,0:07:12.42,0:07:13.40,Default,,0000,0000,0000,,So there you go.
Dialogue: 0,0:07:13.40,0:07:16.12,Default,,0000,0000,0000,,This is a significant\Nmoment in your life.
Dialogue: 0,0:07:16.12,0:07:21.19,Default,,0000,0000,0000,,You have just been exposed to\Nyour first Laplace Transform.
Dialogue: 0,0:07:21.19,0:07:23.35,Default,,0000,0000,0000,,I'll show you in a few videos,\Nthere are whole tables of
Dialogue: 0,0:07:23.35,0:07:25.30,Default,,0000,0000,0000,,Laplace Transforms, and\Neventually we'll
Dialogue: 0,0:07:25.30,0:07:27.57,Default,,0000,0000,0000,,prove all of them.
Dialogue: 0,0:07:27.57,0:07:29.44,Default,,0000,0000,0000,,But for now, we'll just\Nwork through some of
Dialogue: 0,0:07:29.44,0:07:30.23,Default,,0000,0000,0000,,the more basic ones.
Dialogue: 0,0:07:30.23,0:07:32.18,Default,,0000,0000,0000,,But this can be our\Nfirst entry in our
Dialogue: 0,0:07:32.18,0:07:34.68,Default,,0000,0000,0000,,Laplace Transform table.
Dialogue: 0,0:07:34.68,0:07:39.87,Default,,0000,0000,0000,,The Laplace Transform of\Nf of t is equal to
Dialogue: 0,0:07:39.87,0:07:44.03,Default,,0000,0000,0000,,1 is equal to 1/s.
Dialogue: 0,0:07:44.03,0:07:46.43,Default,,0000,0000,0000,,Notice we went from a function\Nof t-- although obviously this
Dialogue: 0,0:07:46.43,0:07:50.46,Default,,0000,0000,0000,,one wasn't really dependent\Non t-- to a function of s.
Dialogue: 0,0:07:50.46,0:07:53.52,Default,,0000,0000,0000,,I have about 3 minutes left,\Nbut I don't think that's
Dialogue: 0,0:07:53.52,0:07:56.01,Default,,0000,0000,0000,,enough time to do another\NLaplace Transform.
Dialogue: 0,0:07:56.01,0:07:59.04,Default,,0000,0000,0000,,So I will save that for\Nthe next video.
Dialogue: 0,0:07:59.04,0:08:00.66,Default,,0000,0000,0000,,See you soon.