so welcome to a series of lectures on mathematical methods and its applications we were discussing
laplace transform and their properties so in this lecture we will see that what are
the initial and final value theorems on laplace transforms and how they are important to find
some laplace transforms of some functions ok now what are what are what is initial value
theorem let us see let laplace transform of f t is suppose f p and f dash t be piecewise
continuous and is of exponential order for t non negative t greater than equal to zero
ok if limit p tends to infinity p f p exists
then this theorem is states that limit p tend to infinity p f p is nothing but limit
t tend into zero f t so this is initial value theorem now what is the proof of this theorem
let us see so what is what is laplace of f dash t it is nothing but we already know
that it is nothing but p f p minus f zero laplace f dash t is nothing but p f p minus
f zero where where this f p is nothing but laplace of f t ok so this f p is nothing but
laplace of f t now let us compute laplace of f dash t by the definition and take
limit p tend in to infinity because in initial value theorem we have to take limit p tend
into infinity ok so what is laplace of f dash t by the main definition of laplace it is
nothing but zero to infinity e k power minus p t f dash t into d t ok and this is which
is equal to by this it is equals to p f p minus f zero
so take limit p tend to infinity both the sides so it is limit p tend to infinity of
zero to infinity e k power minus p t f dash t d t will be equal to limit p tend into infinity
p f p minus f zero now let us compute this part let us compute this part now f dash t
is of exponential order and is piecewise continuous so mode of f dash t will be
nothing but less than equals to some m into e k power m alpha t there will exists
some m and alpha such that mode of f dash t will be less than equal to m into e k power
alpha t since it is exponential order alpha so
we can easily write that this value the left hand side ok the left hand side zero to infinity
e k power minus p t f dash t d t and its modulus will be less than or equals to zero to infinity
e k power minus p t mode f dash t into d t and this mode f dash t less than equal to
this because it is of exponential order of alpha so it is again less than equals to zero
to infinity m into e k power minus p t into e k power alpha t d t and this quantity is
nothing but m into integral zero to infinity e k power minus p minus alpha t into d t so
this this integral is nothing but this integral is nothing but it is m into e k power minus
p minus alpha times t upon minus p minus alpha times t zero to infinity
so when p is greater than alpha so this value is nothing but one m upon p minus alpha because
when p greater than alpha and t tend to infinity it will tends to zero and at zero this will
tends to one so this will nothing but m upon p minus alpha so basically this expression
this expression is less than equal to this expression ok and when p tending to infinity
so this will definitely tend to zero so hence this tends to zero ok because because this
expression is less than this expression less than equal to this expression m upon p
minus alpha and as p tending to infinity this expression will be less than equals to limit
t tend to infinity this expression and as p tend to infinity this is tend in to zero
which is tend in to zero as p tend in to infinity so hence we can say that limit p tending
to infinity zero to infinity e k power minus p t f dash t d t is equal to zero
now using this fact in this expression what you will obtain so if you use this fact over
here so this is nothing but zero ok we have just proved that this expression is nothing
but zero so so from here we will obtain that limit t tend into infinity p f p is nothing
but f zero now when we proved this property of laplace transform laplace transform of
derivatives that laplace transform of f dash t is nothing but this expression so there
we are assume that function is continuous for all t greater than equal to zero ok so
its function is not continuous thats why we are obtaining f zero over here because we
are assuming f is continuous throughout non negative side of real axis ok
so thats why we obtain f zero over here if it if function is not continuous t equal
to zero then instead of obtaining f zero we will obtain limit t tending to zero f zero
over here f t over here instead of this expression over here we will obtain this expression if
f is not continuous at t equal to zero ok so so basically so basically things are
same if [finc/function] function is continuous at t equal to zero so this is nothing but
f zero however a function is not continuous at t equal to zero so this is nothing but
this expression nothing but f zero plus on the right hand side of t equal to zero
ok it is a right hand limit of t equal to zero that is f zero plus
so therefore to generalize this theorem we are writing on the right hand side we are
writing t tending to zero f t because if function is continuous so right hand side will definitely
be f zero if function is not continuous at t equal to zero so right hand side is nothing
but f zero plus so in both the case it will work ok so thats why we are writing limit
t tend into zero f t in the right hand side instead of writing f zero ok so this is initial
value theorem now now we will come to final value theorem now what final value theorem
state let us see if laplace transform of f t is f p suppose and f dash t is piecewise
continuous for t greater than equal to zero and if limit tend in to infinity f t exists
then this result hold or now what this result is so this was initial value theorem
initial value theorem and what what is final value theorem it is limit p tend into to zero
f p f p is nothing but limit t tend in to infinity f t so this is final value theorem
so let us now prove the final value theorem how we will obtain final value theorem let
us see now again again we will go to the same expression
what is that limit laplace transform of f dash t is nothing but p f p minus f zero
so again this f zero will be limit t tending to zero f t if function is not continuous
at t equal to zero ok and this this is this laplace of f dash t is nothing but from the
definition of laplace transforms zero to infinity e k power minus p t f dash t into d t now
we want now we want limit p tend into zero ok this side so same here same on this side
also so this is equal to p f p minus f zero here we are assuming f is continuous
at t equal to zero take limit p tend in to zero both the sides
so what will obtain limit p tend in to zero zero to infinity e k power minus p t f dash
t d t and the right hand side will remain the same limit p tend in to zero p f p minus
f zero ok so now if p tend into zero this will tend to one so this side is nothing but
this implies zero to infinity f dash t d t which is equals to limit p tend in to zero
p f p minus f zero now let us see the left hand side what is left hand side left hand
side is zero to infinity f dash t d t this is right left hand side so what this expression
is basically this is something limit k tend into infinity zero to k something f dash
t d t now since derivative integral is anti derivative we already know this so this is
nothing but limit k tending to infinity f t from zero to k
so this is nothing but limit k tend into infinity f k minus f zero ok and which is nothing but
we can simply write limit lim t tend in to infinity f t minus f zero this we can simply
write like this so this is left hand side so left hand side is left hand side is equal
to this expression so when we equate both the sides so this will be this will be
equal to from this side is nothing but limit p tend into zero p f p minus f zero so f zero
f zero cut cancels out both the sides so this we obtain this result which is final value
theorem ok if function is continuous at t equal to zero we are having f zero over
here if it is not continuous at t equal to zero we will be having f zero plus here also
here also so in both the cases whether function is continuous
or not equal to zero f zero or f zero plus will cancel from both the sides ok so finally
we will be having this expression which you which we are calling as final value theorem
ok now let us see some of the problem based on initial final value theorem first of
all let us see the simple example of f t equal to two e k power minus six t
let us illustrate this using these two theorems so one can easily illustrate this very
easily you see that first is suppose initial value theorem suppose first is initial value
theorem initial value theorem first suppose you want to illustrate this theorem using
this example for this example sorry so what is f t f t is two e k power minus six t so
what is laplace of f t very simple it is two upon p plus six so this we are calling as
f p now see initial value theorem the left hand side of initial value theorem is limit
p tending to infinity p f p so what is limit tend into infinity p f p
p into f p that is two p upon p plus six and as p tends to infinity this will tends to
it divide numerator denominator by p this is this quantity and this will tends to two
now the left hand side limit t tending to zero f t so limit t tend into zero what
is f t to e k power minus six t so it is nothing but two so hence initial value theorem has
been verified because left hand side is equal to right hand side ok now come to final value
theorem for the same problem so now the final value theorem so what is left hand side
for final value theorem limit p tend into zero p f p
so it is limit p tend into zero p f p is two p upon p plus six and it will is equal to
zero equal to zero now what is limit t tend in to infinity f t it is limit t tend in to
infinity f t is two e k power minus six t which is obviously zero because at t tends
to infinity e k power minus t tends to zero so that will tends to zero so hence since
left hand side equal to right hand side hence the final value theorem has been verified
so the first problem has been solved now come to second problem given at f p is this ok
find f zero so we already know f p which is laplace transform of some f t ok now without
finding laplace inverse we can easily find what is f zero how using initial value theorem
so here here we are assuming that function is continuous at t equal to zero
ok thats then then only we can obtain f zero ok so what is what is f p for this problem
what is f p given to us it is p plus three upon p plus two whole square plus three square
this is f p now what will be what is f zero f zero is nothing but by the initial value
theorem is f is continuous at t equal to zero f zero is nothing but limit p tending to infinity
p f p so it is nothing but limit p tending to infinity p into f p p into f p
so this you can easily see that this limit will tends to one so it is equal to one so
what is a value of f zero this one so without finding laplace inverse of this function we
can easily find out what is the value of f zero using initial value theorem ok similarly
if you want to find out limit t tend to infinity f t so that also we can find out using final
value theorem so limit t tend in to infinity f t will be
nothing but using final value theorem it is equal to this expression limit p tend in to
zero p f p and it is nothing but limit p tend into zero p in to f p is p plus three upon
p plus two whole square plus nine and it is nothing but is equal to zero so limit t tend
to infinity f t is nothing but zero so that also we can find out using initial and
final value theorems now let us solve some problems which we are already discussed
in the previous lectures that how to find laplace transform of these functions but let
us find these function using initial value theorem let us [sim/solve] solve first problem
ok so i am telling these problem so are here because these theorems will play an
important role while we solve some deferential equations so thats why we must have information
about what initial final value theorem are and how they are important and how we can
solve problem based on this ok so what how to find laplace transform of this
function zero to t it is sin u upon u d u so so we already see that how to find laplace
of this we first find laplace of sin u sin t which is one upon p square plus one [FL]
sin t upon t is giver by integral p to infinity f f p d p ok and then for zero to
t it is nothing but f p by p which is f p is laplace of this entire function so that
we have already discussed in the last lectures now how to find laplace of these functions
using initial value theorem ok so let us see now let us let us call this whole function
as f t so f t is equals to zero to t sin u upon u d u so what is f dash u derivative
of f respect to t that we can find out using leibniz theorem so what will be the derivative
of this it will be sin t by t into one minus zero ok so that is quite obvious that this
is will be this and of course one can easily see that f zero is nothing but zero
because when you take t tend into zero or t equal to zero so this lower and upper
limit both are same value will be zero ok now here t f dash t is nothing but sin t t
f dash t is nothing but sin t ok take laplace both the sides so laplace of t f dash t is
nothing but laplace of sin t ok where interested to find out laplace of f t ok we have to find
out laplace of f t using initial value theorem ok now laplace of t in to some function is
nothing but we already know that laplace of t k power n in to some function say f t
it is nothing but minus one k power n d by d p of f p where f p is nothing but laplace
transform n th derivative sorry n th derivative of f p where f p is nothing but laplace of
this f t now here in place of f t we have f dash t
so what will be this expression will be minus one k power one n is one so it is d by d p
of laplace of f dash t and what is laplace of sin t it is one upon p square plus one
now this is further equal to minus of d by d p of laplace of f dash t is laplace of this
t is p f p minus f zero is equals to one upon p square plus one ok so this expression further
can be written as minus d by d p of p f p is equal to one upon p square plus one derivative
of this this respect to the p will be zero its a constant quantity to derivative will
be zero now it is nothing but minus p f p is equals to take integration both the sides
respect to p so it is tends to inverse p plus some arbitrary constant c because it is an
indefinite integral ok now how to find c so to find c take limit p tending to infinity
both the sides ok limit p tend in to infinity n both the sides
so it is limit p tend to infinity minus p f p it is limit p tend in to infinity tan
inverse p plus c now this quantity is by the initial value theorem is nothing but limit
t tend to zero f t and limit t tend to zero f t which is f zero here is nothing but zero
ok which is nothing but zero so we can say that this is zero because this quan[tity]
this expression p f p limit t tend into infinity is nothing but t tend to zero f t and limit
t is tend to zero f t is f zero which is zero ok so this is zero and this is pi by two plus
c so c is nothing but minus pi by two so what will be f p from here so f p will be nothing
but minus one by p times tan inverse p minus pi by two so and what is f p is laplace transform
of f t so hence laplace transform of this f t is nothing but this
so basically initial final value theorem have important to find out this arbitrary constant
c otherwise it is difficult to find ok hence this laplace transform of this f t is nothing
but this expression now let us compute the second problem this problem can be solved
also using initial value theorem e k power it is u sin square u upon cos upon u sorry
upon u into d u so again you will take this function as say f t and try to find out
it laplace transform using initial value theorem on the same lines we did earlier so take this
is f t what is f t f t is zero to t e k power u into sin square u upon u into d u
ok so what is f dash t is e k power t sin square t by t by leibniz theorem and of course
limit t tend to zero f t or we are calling as f zero is nothing but zero ok so take
t in the right hand left hand side so it is this expression take laplace transform
both the sides laplace transform of left hand side is minus
one k power one d by d p of laplace transform of f dash t now to find laplace transform
of this expression first in laplace transform of sin square t and then apply first shifting
property so what is laplace transform of sin square t laplace transform of sin square t
this we can finding out laplace transform of one minus cos two t by two and it is nothing
but one by two p minus one by two in to p upon p square plus four so it is nothing but
and again apply shifting property so you replace p by p minus one so it is one upon
two times one upon p minus one minus p minus one upon p minus one whole square plus four
ok you replace p by p minus one now minus of d by d p of it is p f p minus
f zero which is one by two of one upon p minus one minus p minus one upon p minus one whole
square plus four derivative of this respect to p zero so it is minus p f p you integrate
both sides respect to p and derivative of this is zero ok so this is one by two comes
out this is log of p minus one and multiply and divide by two so it is minus one by two
log of p minus one whole square plus four so from here f p is nothing but and plus c
also because it is an indefinite integral ok so f p is nothing but minus one by two
p ok take p over here only and one by two it is log p minus one upon under root p minus
one whole square plus four and plus c now you can use initial value theorem to find
out the value of c take limit p tending to infinity both the sides as p tend to infinity
this side will tends to this side will be equal to limit t tending to zero f t which
is nothing but zero so this side will be zero as p tend to infinity this will tends to one
and log one is zero so c is nothing but zero so c will be zero ok and f p can be find out
the f p will be nothing but you divide this by minus p so f and f p is nothing but
laplace transform of this f t so hence we can find out laplace transform of this function
so similarly the third problem can also be solved using the same concept and initial
value theorem so thank you very much