WEBVTT
Kind: captions
Language: en
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yeah hello everyone so in the last lecture
we were studying about properties of laplace
00:00:24.820 --> 00:00:30.939
transform some of the properties we have seen
like shifting property first shifting or
00:00:30.939 --> 00:00:36.160
first translation or shifting property then
we see second translation or shifting property
00:00:36.160 --> 00:00:42.180
some problems based on this that we have seen
in the last lecture so now we will see some
00:00:42.180 --> 00:00:50.210
more prosperities of laplace transforms in
this lecture so the next property is change
00:00:50.210 --> 00:00:58.530
of scale property now what does states it
states that if laplace transform of some f
00:00:58.530 --> 00:01:17.250
t is f t then laplace transform of f a t is
nothing but one by a f p by a of course a
00:01:17.250 --> 00:01:24.359
should not equal to zero so this is called
change of scale property
00:01:24.359 --> 00:01:30.689
now how we will how we it we will prove
it so the proof is very simple see simply
00:01:30.689 --> 00:01:39.020
based on the definition of laplace transforms
so what is laplace transform of f a t it is
00:01:39.020 --> 00:01:49.759
nothing but zero to infinity e key power minus
p t f a t now you can substitute a t equal
00:01:49.759 --> 00:01:56.329
to some new variable z so a d t will be equals
to d z so where you substitute this thing
00:01:56.329 --> 00:02:05.399
over here so when t is zero z is zero so it
will be zero when t tend into infinity z will
00:02:05.399 --> 00:02:14.180
tend to infinity so it is infinity e key power
now t is nothing but z by a so it is minus
00:02:14.180 --> 00:02:25.530
p into z upon a into f of z and d t is nothing
but d z upon a so that is one upon a times
00:02:25.530 --> 00:02:36.000
d z so this quantity is nothing but one by
a zero to infinity e key power minus p z upon
00:02:36.000 --> 00:02:50.870
a f z into d z or we can write it like this
one upon a integer zero to infinity e key
00:02:50.870 --> 00:03:04.230
power minus some zeta times p into f z d z
where zeta is nothing but p upon a
00:03:04.230 --> 00:03:12.290
now when you compare this with laplace transform
of f t so laplace transform of f t is is given
00:03:12.290 --> 00:03:23.650
by a zero to infinity e key power minus p
t f t d t and this we are calling as f p this
00:03:23.650 --> 00:03:31.210
we are calling as f p because p is a parameter
two is a function of f p now here the same
00:03:31.210 --> 00:03:37.910
thing is same thing what we have in with
this expression so this can also be written
00:03:37.910 --> 00:03:45.819
as this expression can also be written as
zero to infinity one upon a times e key power
00:03:45.819 --> 00:03:59.690
minus zeta t p into f t d t so that is
nothing but one upon a f zeta because instead
00:03:59.690 --> 00:04:13.170
of ok it is it is z because variable is z
so it is z so now z is here so instead of
00:04:13.170 --> 00:04:23.690
it is t now t is here so instead of p we
have zhy here so it is will be f zhy so what
00:04:23.690 --> 00:04:33.770
is zhy it is one by a f of p upon a so that
is a proof of change of scale property
00:04:33.770 --> 00:04:43.419
that is laplace of f a t is nothing but
one by a f of p by a
00:04:43.419 --> 00:04:51.490
now we will see a [pro/some] some problems
based on this suppose it is given to us
00:04:51.490 --> 00:05:01.900
that laplace transform of f t is nothing but
one by p p square plus one and we have to
00:05:01.900 --> 00:05:09.229
find laplace transform of e key power minus
two t into f of three t now there are two
00:05:09.229 --> 00:05:15.620
ways to solve this problem the one way is
you find the laplace inverse of this f p this
00:05:15.620 --> 00:05:25.270
is f p you find the laplace inverse of this
f p that will be nothing but f t find f of
00:05:25.270 --> 00:05:33.800
three t and then f of three t will be find
and then using shifting property you can find
00:05:33.800 --> 00:05:40.960
laplace transform of e key power minus three
t f three t and second way is you apply change
00:05:40.960 --> 00:05:46.900
of scale property and then the shifting property
to directly find out the laplace transform
00:05:46.900 --> 00:05:50.919
of this expression
now how you can find a laplace inverse of
00:05:50.919 --> 00:05:59.830
this again using partial fractions this is
nothing but some a upon p plus b p plus c
00:05:59.830 --> 00:06:12.090
upon p square plus one so one is equals to
a into p square plus one plus b p plus c into
00:06:12.090 --> 00:06:25.210
p so when you take p equal to zero this implies
one is equals to a so a is nothing but
00:06:25.210 --> 00:06:35.560
one now you compare the coefficient of p square
here p square is zero here p square is a plus
00:06:35.560 --> 00:06:44.430
b so this implies b is minus a and hence is
equals to minus one because a is one now for
00:06:44.430 --> 00:06:52.629
c you can compare the coefficient of p here
p coefficient is only this which is c and
00:06:52.629 --> 00:06:59.220
here there is no coefficient involving p so
zero equal to c so c must be zero so what
00:06:59.220 --> 00:07:05.639
are the partial fraction of this expression
one by p into p square plus one the partial
00:07:05.639 --> 00:07:14.509
fraction will be nothing but a is one so it
is one by p plus b is minus one and c is zero
00:07:14.509 --> 00:07:23.610
so it is minus p upon p square plus one so
this ultra partial fraction of this f p so
00:07:23.610 --> 00:07:32.009
what will be laplace inverse of this so by
the linearity property of laplace inverse
00:07:32.009 --> 00:07:39.650
this is nothing but laplace inverse of one
by p minus laplace inverse of p upon p square
00:07:39.650 --> 00:07:50.099
plus one and this is nothing but one minus
cos t so this is f t laplace inverse of f
00:07:50.099 --> 00:07:56.699
p is nothing but f t now we have to find out
first f three t so what will be f three t
00:07:56.699 --> 00:08:08.520
it is nothing but one minus cos three t and
laplace of e key power minus two t into f
00:08:08.520 --> 00:08:19.039
three t will be nothing but first we will
find out laplace transform of f three t
00:08:19.039 --> 00:08:24.689
that will be new f p and in that new f p we
will apply shifting property
00:08:24.689 --> 00:08:34.000
so what is laplace transform of f three t
which is nothing but one by p minus three
00:08:34.000 --> 00:08:43.039
upon p minus p upon sorry p upon p square
plus nine where cos a t is nothing but laplace
00:08:43.039 --> 00:08:50.470
of cos a t is p upon p square plus a square
so laplace of so this is this we are taking
00:08:50.470 --> 00:08:58.510
as suppose g p function of p suppose it is
g p so laplace of f three t is g p then laplace
00:08:58.510 --> 00:09:07.730
of this expression will be nothing but g you
replace p by a p minus a here a is minus
00:09:07.730 --> 00:09:15.079
two so it is nothing but p plus two so that
is nothing but one upon p plus two minus p
00:09:15.079 --> 00:09:25.850
plus two upon p plus two whole square plus
nine so if you take l c m and simplify it
00:09:25.850 --> 00:09:33.320
is clearly [vis/visible] visible that it
is nothing but p plus two into p plus two
00:09:33.320 --> 00:09:42.269
whole square plus nine so that will be the
laplace of e key power minus three t into
00:09:42.269 --> 00:09:47.019
f three t e key power minus two t into f three
t
00:09:47.019 --> 00:09:55.420
so this is the one way now finding laplace
inverse of given f p this f p is very
00:09:55.420 --> 00:10:00.220
simple so we have taken the partial fraction
and simply computed first computed laplace
00:10:00.220 --> 00:10:07.280
inverse of this f p let it suppose it is f
t and then using shifting property we have
00:10:07.280 --> 00:10:13.420
find the laplace transform of e key power
minus two t into f three t now if you have
00:10:13.420 --> 00:10:20.310
complicated expression so finding laplace
inverse is a difficult job i mean is
00:10:20.310 --> 00:10:26.720
a time consuming job so for this particular
type of problems we can solve directly also
00:10:26.720 --> 00:10:33.639
using first change of scale property and then
using shifting property how now let us
00:10:33.639 --> 00:10:46.220
see so this this is suppose f p ok this is
suppose now what is laplace of f three t it
00:10:46.220 --> 00:10:56.700
is nothing but one by a f a here is three
so one by three f of p by three so it is nothing
00:10:56.700 --> 00:11:06.110
but one by three one upon p by three into
p square by nine plus one so that is nothing
00:11:06.110 --> 00:11:12.440
but three three cancels out ok this is p by
three and this is one by three ok this is
00:11:12.440 --> 00:11:20.160
one by three
so this is one by three into so this is
00:11:20.160 --> 00:11:29.339
three into nine upon p into p square plus
nine so this three three cancels out now
00:11:29.339 --> 00:11:39.650
what is laplace of e key power minus two t
f three t so this this suppose this is some
00:11:39.650 --> 00:11:46.630
g p function of p suppose this is g p so laplace
of e key power a t or in some f t will be
00:11:46.630 --> 00:11:54.540
nothing but by a shifting property it is
f of p minus a here laplace of this function
00:11:54.540 --> 00:12:02.990
is g p so we have to apply shifting property
for this function so this will be nothing
00:12:02.990 --> 00:12:10.730
but by the shifting property it is p plus
two because here a is minus two so it is nothing
00:12:10.730 --> 00:12:22.230
but nine upon p plus two into p plus two whole
square plus nine so that is how using change
00:12:22.230 --> 00:12:27.990
of scale property and shifting property we
can find the laplace transform of this particular
00:12:27.990 --> 00:12:39.100
problem very easily ok now let us find out
laplace transforms of some special functions
00:12:39.100 --> 00:12:45.639
special functions like bessel function
so i am not going into much detail of special
00:12:45.639 --> 00:12:55.050
functions simply assume that j naught x
is given by this series and error function
00:12:55.050 --> 00:13:02.040
all is given by this particular expression
so how to find laplace transform of j naught
00:13:02.040 --> 00:13:08.320
t which we call as bassel function and how
to find laplace transform of error function
00:13:08.320 --> 00:13:15.550
so let us try to find out this thing and
then once we find laplace transform of bessels
00:13:15.550 --> 00:13:23.120
function which is j naught t or error function
we can solve those two problems using shifting
00:13:23.120 --> 00:13:30.690
property and change of scale property so let
us find first j naught t so j naught t is
00:13:30.690 --> 00:13:41.430
nothing but one minus t square by two square
plus t key power four upon two square into
00:13:41.430 --> 00:13:47.779
four square minus t key power six upon two
square four square into six square and so
00:13:47.779 --> 00:13:56.040
on as given in the problem ok
so what is laplace transform of j naught t
00:13:56.040 --> 00:14:09.860
now laplace transform one is one by p laplace
transform of t square is factorial two upon
00:14:09.860 --> 00:14:19.310
p square into two square laplace transform
p q it is because laplace transform of t key
00:14:19.310 --> 00:14:28.070
power n is what is nothing but factorial n
upon p key power n plus one if n is one two
00:14:28.070 --> 00:14:36.720
three and so on for integer for positive integers
so again it is factorial four upon p key power
00:14:36.720 --> 00:14:45.990
five two square into four square minus factorial
six upon p key power seven two square four
00:14:45.990 --> 00:14:55.750
square six square and so on so it is nothing
but one by p take one by p common so it is
00:14:55.750 --> 00:15:05.269
one minus one two two cancels out so it is
one by two p square then it is nothing but
00:15:05.269 --> 00:15:15.959
four into three into two into one upon p key
power four into two square into four square
00:15:15.959 --> 00:15:22.880
minus it is nothing but six into five into
four into three into two into one upon p key
00:15:22.880 --> 00:15:28.399
power seven p key power six because one
p is common
00:15:28.399 --> 00:15:40.019
so and it is two square four square and six
square and so on now one two two cancel out
00:15:40.019 --> 00:15:51.230
one four four cancel out so we have observed
that numerator always contain odd terms and
00:15:51.230 --> 00:15:59.370
denominator always contain even terms that
that is it is nothing but one by p minus one
00:15:59.370 --> 00:16:07.839
minus one by two p square plus one into three
upon two into four into one by p square ka
00:16:07.839 --> 00:16:17.529
whole square it is nothing but minus one into
three into five upon two into four into six
00:16:17.529 --> 00:16:26.040
into one by p square ka whole cube and so
on
00:16:26.040 --> 00:16:33.529
ok so this is nothing but one by p one by
p into meh this is one by p into one by p
00:16:33.529 --> 00:16:46.390
into now this is one minus this is nothing
but you can say plus minus one by two minus
00:16:46.390 --> 00:16:53.510
one by two into p one by p square now this
is nothing but minus one by two into two minus
00:16:53.510 --> 00:16:59.420
one by two minus one upon factorial two into
one by p square when you simplify this expression
00:16:59.420 --> 00:17:06.780
so what we get it is minus one into minus
three when you taken l c m so one into three
00:17:06.780 --> 00:17:12.079
minus minus plus and two cube so two cube
in the denominator
00:17:12.079 --> 00:17:20.150
ok next is it is plus minus one by two
minus one by two minus one into minus one
00:17:20.150 --> 00:17:30.630
by two minus two upon factorial three into
one by p square ka whole cube so again it
00:17:30.630 --> 00:17:35.820
is nothing but when you simplify so you will
get back the same expression it is one into
00:17:35.820 --> 00:17:41.300
three into five upon two cube into factorial
three which is nothing but two into four into
00:17:41.300 --> 00:17:48.720
six so what this expression basically this
is nothing but one minus one by p sq[uare]
00:17:48.720 --> 00:17:56.020
one plus p square key power minus half this
is this is by the binomial theorem one plus
00:17:56.020 --> 00:18:02.500
m x plus m minus upon factorial to into x
square and so on so now this is nothing but
00:18:02.500 --> 00:18:09.470
when you simplify it this is one by p into
p square plus one upon p square whole power
00:18:09.470 --> 00:18:18.910
minus half which is nothing but one upon p
square plus one ka under root so this is the
00:18:18.910 --> 00:18:27.360
laplace transform of j naught t which we are
calling as bessel function ok so you simply
00:18:27.360 --> 00:18:32.090
open the series this is the series is given
to you you take the laplace transform both
00:18:32.090 --> 00:18:37.050
the side simplify
so this will be reduce to some binomial expansion
00:18:37.050 --> 00:18:42.550
of one plus one by p square whole key power
minus half and when you simplify we will get
00:18:42.550 --> 00:18:48.870
one upon under root p square plus one at
the laplace of laplace transform of j naught
00:18:48.870 --> 00:18:57.250
t now now we can solve first problem since
now we know laplace transform of j naught
00:18:57.250 --> 00:19:09.020
t so what will be the laplace transform of
j naught two t first so this is using
00:19:09.020 --> 00:19:15.840
change of scale property because if laplace
transform f t is some f p then laplace transform
00:19:15.840 --> 00:19:24.320
f a t is nothing but is nothing but one
by a f p by a so here here here this is f
00:19:24.320 --> 00:19:32.060
p and this is this is laplace transform of
j naught t so what will be laplace transform
00:19:32.060 --> 00:19:42.210
of this here it will be one by two f p by
two so this nothing but one by two f of one
00:19:42.210 --> 00:19:52.340
by under root p by two ka whole square plus
one which is nothing but two upon two under
00:19:52.340 --> 00:19:57.300
root p square plus one p square plus four
so it is nothing but two two cancel out
00:19:57.300 --> 00:20:03.820
so it is one by under root p square plus four
so what should be laplace transform of e key
00:20:03.820 --> 00:20:15.030
power minus three t of this j naught two t
this will be equal to now let us suppose this
00:20:15.030 --> 00:20:23.370
is g p the laplace transform of j naught two
t there a suppose it is g p so laplace transform
00:20:23.370 --> 00:20:28.160
of e key power minus three t into lap[lace]
into this is nothing but by using shifting
00:20:28.160 --> 00:20:36.820
property it is g of p plus three because in
the laplace transform this j naught two t
00:20:36.820 --> 00:20:45.549
you replace p by p plus three so laplace transform
of this is g p so you replace p by p minus
00:20:45.549 --> 00:20:53.530
a and a is minus three so it is nothing but
one upon under root p plus three whole square
00:20:53.530 --> 00:21:02.300
plus four so this will be the final answer
you can simplify and that should be the laplace
00:21:02.300 --> 00:21:08.480
transform of e key power minus three t j naught
two t now let us find out laplace transform
00:21:08.480 --> 00:21:13.430
of error function under root t
so and then using change of scale property
00:21:13.430 --> 00:21:19.680
we can solve that problem so what is laplace
transform of error function let us see so
00:21:19.680 --> 00:21:27.820
error function what is error function of under
root t how you define it it is nothing but
00:21:27.820 --> 00:21:39.550
zero to under root t two upon pi times zero
to under root t e u square e key power minus
00:21:39.550 --> 00:21:49.520
u square d u so now it is equal to two upon
under root pi zero to under root t now
00:21:49.520 --> 00:21:55.380
we know the expression of e key power minus
x we can use this expression here it is one
00:21:55.380 --> 00:22:04.170
minus u square upon factorial one mi[nus]
plus u square ka whole square upon factorial
00:22:04.170 --> 00:22:15.770
two minus u square ka whole square upon factorial
three and so on into d u so this is nothing
00:22:15.770 --> 00:22:24.160
but two upon under root pi so integral of
one will be u and when you apply a upper limit
00:22:24.160 --> 00:22:30.770
upper limit is sorry it is under root t
ok so when you apply upper limit minus lower
00:22:30.770 --> 00:22:37.680
limit
so this will be nothing but under root t minus
00:22:37.680 --> 00:22:45.250
now integration of u square is u cube upon
three so u cube and when you apply upper limit
00:22:45.250 --> 00:22:51.730
minus lower limit is nothing but t key power
three by two upon three into factorial one
00:22:51.730 --> 00:22:57.750
plus u key power four ka integral is u key
power five upon five and when you apply the
00:22:57.750 --> 00:23:08.420
limit is t key power five by two upon five
factorial two similarly here it will be
00:23:08.420 --> 00:23:16.860
u key power six that is t key power seven
by two upon seven factorial three and so on
00:23:16.860 --> 00:23:24.230
so this would be the expression of error
function in the series from now when you take
00:23:24.230 --> 00:23:29.770
laplace transform of e key error function
of under root t so this will be nothing but
00:23:29.770 --> 00:23:35.290
two upon under root pi you take common and
this is nothing but gamma three by two upon
00:23:35.290 --> 00:23:44.480
three by two upon p key power three by two
minus it is gamma five by two upon three into
00:23:44.480 --> 00:23:57.030
factorial one into p key power five by two
ok plus gamma seven by two upon five into
00:23:57.030 --> 00:24:05.050
two factorial into p key power seven by two
and so on this is nothing
00:24:05.050 --> 00:24:11.480
but again two upon under root pi take gamma
three by two common and p key power three
00:24:11.480 --> 00:24:16.580
by two common from the entire expression from
the entire series now gamma three by two is
00:24:16.580 --> 00:24:24.520
nothing but one by two under root pi this
we know as gamma and plus one is n gamma n
00:24:24.520 --> 00:24:31.480
so this is one by two under root pi this i
have taken common and also i have taken p
00:24:31.480 --> 00:24:39.730
key power three by two as common this nothing
but one minus now gamma five by two is
00:24:39.730 --> 00:24:47.490
three by two gamma three by two gamma three
by two already common so it is three by two
00:24:47.490 --> 00:24:57.360
into three into factorial one and p key power
three by two is common so it is p ok plus
00:24:57.360 --> 00:25:02.850
now gamma seven by two is five by two gamma
five by two and gamma five by two can be [ag/again]
00:25:02.850 --> 00:25:06.180
again written as three by two gamma three
by two
00:25:06.180 --> 00:25:14.140
so it is nothing but five by two into three
by two gamma three by two which is common
00:25:14.140 --> 00:25:26.730
and upon five into factorial two p key power
two and so on now again let us simplify this
00:25:26.730 --> 00:25:32.110
expression two two cancels out this is cancels
out one upon p key power three by two and
00:25:32.110 --> 00:25:39.930
it is one minus this three three cancel out
so it is one minus one by two p plus this
00:25:39.930 --> 00:25:46.740
five five cancels out it is nothing but one
into three upon two into four upon one by
00:25:46.740 --> 00:25:56.280
p ka whole square and so on so it is something
the same expression which we arrived in
00:25:56.280 --> 00:26:02.790
the derivation of jacobian derivation of
sorry bessel function j naught t so it
00:26:02.790 --> 00:26:10.910
is something like that so we can also write
this as one upon p key power three by two
00:26:10.910 --> 00:26:16.990
one remain one one minus this so this can
be written as minus one by two minus one by
00:26:16.990 --> 00:26:23.380
two minus one upon factorial two into one
by p square and so on
00:26:23.380 --> 00:26:32.260
similarly the other expression can be write
in the same way so it is nothing but it is
00:26:32.260 --> 00:26:39.390
nothing but one upon p key power three by
two one min[us] one plus one by p whole
00:26:39.390 --> 00:26:46.780
key power minus half in that expression
we have a binomial series of one plus one
00:26:46.780 --> 00:26:51.470
by p square key power minus half here we have
a expre[ssion] binomial series of one plus
00:26:51.470 --> 00:26:56.670
one by p key power minus half so when you
simplify it so it is nothing but one upon
00:26:56.670 --> 00:27:06.600
p key power three by two into into p key
power half upon under root t plus one so it
00:27:06.600 --> 00:27:14.550
is nothing but one upon p under root p plus
one so this is the laplace transform of error
00:27:14.550 --> 00:27:20.610
function of under root t this is how we can
find out laplace transform of error function
00:27:20.610 --> 00:27:26.770
of under root t ok now we can solve that
problem very easily
00:27:26.770 --> 00:27:32.920
because we know the laplace transform under
root t which is given by this f p now laplace
00:27:32.920 --> 00:27:38.480
transform of two under root t we can use
change of scale property it will be one by
00:27:38.480 --> 00:27:49.300
two f p by two hm if it is f p if it is f
p so laplace transform of error function of
00:27:49.300 --> 00:27:57.410
two under root t will be nothing but one by
two f p by two this is by the scale property
00:27:57.410 --> 00:28:03.790
change of scale property so we can use this
so it is one by two p by two under root p
00:28:03.790 --> 00:28:12.660
by two plus one so this will be the laplace
transform of that expression
00:28:12.660 --> 00:28:18.600
now the next problem let us first find
laplace transform l n x what is l n x l n
00:28:18.600 --> 00:28:27.250
x is luxurious polynomial ok so we are
not going into the detail of this [bus/but]
00:28:27.250 --> 00:28:32.320
l n x is given by this particular expression
this expression this expression is called
00:28:32.320 --> 00:28:37.890
as l n x let us try to find out the laplace
transform of this and again using shifting
00:28:37.890 --> 00:28:44.290
property or scaling property we can find
out the laplace transform of this expression
00:28:44.290 --> 00:28:48.600
so let us try to find out laplace transform
of l n x
00:28:48.600 --> 00:28:54.770
then the rest of the part you can easily solve
yourself that how can you find out laplace
00:28:54.770 --> 00:29:04.960
transform of e key power two minus two
t l n three t using shifting or scaling
00:29:04.960 --> 00:29:20.920
property
so what is l n x l n x is given by it is e
00:29:20.920 --> 00:29:34.390
key power x upon factorial n n th derivative
of n th derivative of x key power n e key
00:29:34.390 --> 00:29:45.100
power minus x so laplace transform of l n
x by the definition of laplace transform is
00:29:45.100 --> 00:29:53.260
nothing but zero to infinity e key power minus
p t e key power minus p t into e key power
00:29:53.260 --> 00:30:08.150
x upon factorial n n x derivative of this
into x key power n e key power minus x
00:30:08.150 --> 00:30:14.720
so we have to use one symbol because here
t is involved so we have to define this in
00:30:14.720 --> 00:30:30.570
terms of t l n t because laplace transform
f t we are defining
00:30:30.570 --> 00:30:37.190
so we have to express l n t and terms of t
so this is the expression for l n x so we
00:30:37.190 --> 00:30:43.780
can reduce in terms of t replace x by t so
how can you find out laplace transform of
00:30:43.780 --> 00:30:49.630
this function so this is nothing but one by
factorial n can be taken out so it is zero
00:30:49.630 --> 00:30:58.460
to infinity e key power minus p minus one
times t into n th derivative respect to t
00:30:58.460 --> 00:31:08.150
t key power n e key power minus t into d t
now you can make use of integration by
00:31:08.150 --> 00:31:20.770
paths so let us suppose this first function
and the second function so first as it is
00:31:20.770 --> 00:31:30.320
integration of second it is n minus one derivative
now
00:31:30.320 --> 00:31:44.270
integration zero to infinity of this expression
minus integration derivative of first integration
00:31:44.270 --> 00:32:03.400
of second because integration is anti derivative
so derivative powers [will] reduce now
00:32:03.400 --> 00:32:08.450
when t tends to infinity because this have
negative powers and we are assuming p greater
00:32:08.450 --> 00:32:14.530
than one we can assume so this will tends
to zero [to/so] and hence entire term will
00:32:14.530 --> 00:32:22.820
tend to zero and when t tend to zero this
is this this is one and since it has a in
00:32:22.820 --> 00:32:29.210
the n th n minus one derivative of this function
all the terms involved at least one power
00:32:29.210 --> 00:32:36.190
of t because here the highest power of t is
t key power n and when we differentiate n
00:32:36.190 --> 00:32:43.190
minus one times of this aporta function so
all the terms will involve at least t key
00:32:43.190 --> 00:32:49.510
power one so when you take t equal to zero
that value will be zero so this is one by
00:32:49.510 --> 00:32:59.440
factorial n into zero minus now minus minus
plus so it is p minus one times integer zero
00:32:59.440 --> 00:33:08.450
to infinity e key power minus p minus one
times t n minus one derivative of this expression
00:33:08.450 --> 00:33:11.550
into t key power n e key power minus t into
d t
00:33:11.550 --> 00:33:24.920
now now t key yeah it is d t sorry it is
d t because we are differencing with respect
00:33:24.920 --> 00:33:38.970
to t now so it is d t so so we have observed
that in in the integration of this p minus
00:33:38.970 --> 00:33:45.000
one comes out when we integrate first time
one times and n minus one derivative will
00:33:45.000 --> 00:33:55.940
come here so if we repeat this process n times
so then we will be having if it if integrate
00:33:55.940 --> 00:34:00.110
again for second time this is the first function
this is the second function again we will
00:34:00.110 --> 00:34:05.790
take one one power of p minus one one more
power of p minus one outside the bracket
00:34:05.790 --> 00:34:11.849
so that is p minus one whole square will be
here now when we when we integrate this expression
00:34:11.849 --> 00:34:20.139
n times so hm this will be nothing but power
n and it will be involve zero to infinity
00:34:20.139 --> 00:34:25.919
e key power minus p minus one times t and
there will be no derivative of this expression
00:34:25.919 --> 00:34:31.600
[so/to] t key power e key power minus t into
d t
00:34:31.600 --> 00:34:36.990
so this will be nothing but is equals to
one by factorial n this will be nothing but
00:34:36.990 --> 00:34:43.530
now now i write here so this expression will
be nothing but one by factorial n p minus
00:34:43.530 --> 00:34:52.770
n key power n into now e key power t and t
cancels out so this will reduce to laplace
00:34:52.770 --> 00:34:59.660
transform of t key power n and [so/though]
this is equals to one by factorial n p minus
00:34:59.660 --> 00:35:08.820
one key power n and laplace on this is nothing
but factorial n upon p key power n plus one
00:35:08.820 --> 00:35:16.450
because n is a integer so these two cancels
out so hence we have shown that laplace transforms
00:35:16.450 --> 00:35:22.550
l n x is t minus n key power n upon p key
power n plus one now using scale property
00:35:22.550 --> 00:35:32.490
and shifting property we can solve the remaining
part now laplace same property that
00:35:32.490 --> 00:35:37.420
is change of scale property
we can read in terms of inverse also if laplace
00:35:37.420 --> 00:35:43.960
involves of f p of f t then laplace inverse
of p f p by a is nothing but a f a t if a
00:35:43.960 --> 00:35:53.970
is not equal to zero now this [pro/problems]
problems we can easily solve using this
00:35:53.970 --> 00:36:00.700
property suppose first problem we want to
solve so laplace inverse of f p is given to
00:36:00.700 --> 00:36:09.700
us is is e key power t sin under root t
and you want to find out laplace inverse of
00:36:09.700 --> 00:36:17.810
f two p so when you compare with this a is
nothing but one by two a is nothing but one
00:36:17.810 --> 00:36:26.280
by two so you simply replace a by one by two
so that means laplace inverse of f two p will
00:36:26.280 --> 00:36:36.000
be nothing but one by one by two f two by
two because a is half and this expression
00:36:36.000 --> 00:36:42.030
is f t so what should be the laplace inverse
of this expression so laplace inverse of this
00:36:42.030 --> 00:36:52.680
expression will be nothing but nothing
but here a is one by two so it is one by two
00:36:52.680 --> 00:37:02.960
f t by two so it is one by two f t is this
so it is e key power t by two and sin under
00:37:02.960 --> 00:37:12.870
root t by two so using the same property
we can solve the other two problems also its
00:37:12.870 --> 00:37:18.430
very easy you take nine common or four common
and then you can use scaling property to
00:37:18.430 --> 00:37:22.240
solve these two problems also so
thank you very much