Introduction to Differential Equations (includes Order and Degree of a Differential Equation) Video Lecture | Mathematics (Maths) Class 12 - JEE

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what we will study we will study
introduction order and degree of
differential equation general and
particular solutions for differential
equation formation of differential
equation whose general solution and
methods are solving first order and
first degree differential equation so
before we even study differential
equation let's understand why should we
study influential occasion what will
happen to us if we don't see
differentiation the same question which
is generally asking all the videos the
first example of differential equation
is going through a ball from top to down
there we use differential equation if
you talk about population growth we use
differential equation if you talk about
field of science the extensively used it
integration in the field of chemistry
may we talk about reactions we use the
differential equations in Newton's law
of cooling differential equation is the
critical part if you talk about
radioactive decay differential equation
is used if you talk about the circuits
in physics differential equation is used
if you talk about these mixtures House
of mixture a is coming from here and
you're mixing it and it's coming out
this kind of stuff this kind of
real-life problems where you have some
mixture flowing in some mixture flowing
out so in this case also we use
differential equations so before we
understand it actually what is the use
of differential equation all the
scenarios just understand the definition
of differential equation the
differential equation says that there is
nothing but in raishin please note which
defines the relationship between
function and one or more derivative of
the function that means I am trying to
build a relation between function it
assuming this YX be a function or let FX
be a function right and derivative of
the function will be f dash X or F
Double Dash X right by what is f dash X
B by DX
F X what is f double dash X D by DX of F
Phi D d2 by DX 2 y of X right or you can
have F triple dash X so so this is
nothing but a relationship between this
guy that is a function and derivative of
the function right
these guys are my derivative of the
function that means any relations
between this bucket and this bucket will
be my definitely region for example if I
say that I have D by DX of F X this guy
but look k into f ax right plus some
constant this is one differential
equation why because this is the
relationship between derivative of FX FX
and some constant
it may be zero or it's suppose if you
have the your second derivative of this
guy I mean you have this this
relationship with the function itself
plus some constant this is also one
example of differential equation
please note differentially is nothing
but equation where you have a
relationship between relationship
between function and derivatives or the
function so I have derivatives of the
function the function function
derivative of the function in case this
case was first all derivative in this
case we had second order so the
definition is pretty simple equation
where we have relationship between the
function and derivative the function is
called differentiation now let's
understand all the things which we have
studied now the application of
differential equation so we talked about
the ball flowing of the ball falling
Ashley so if you see the ball is falling
in this case we know that by physics
vinnil on the speed with which the ball
is falling will be nothing but in fact
the relation is on the world D H by DT
is some constant that is G
G is the acceleration due to gravity
into T plus one this is the equation we
got from physics actually for this and
the ball falls this is the equation we
get if you have this ball this ball
falls and four does the equation is this
if you see this guy sorry G into H this
is H here G into H so if you see this
guy my H is a function right you can see
here I am I am using derivative of H and
here I am using the H only so I have a
relation between the edge and derivative
of H so this guy is a differential
equation if you talk about population
growth so in this case population grows
if you talk about the change of
population with respect to time is
nothing but some K into P this is how
the published roses because if you have
more population for example India you
have more populations growing faster
contributions less population is it rose
slower that's how it works so the rate
of change of population is also directly
probity the current population so this
is the equation of the publishing growth
and here also if you see we are talking
about this Punk P and derivative of P
there is a relationship between P and
derivative of P so this is also an
example of application of differential
equation in field of science we use a
derivative differentiation a lot for
example in in chemistry if you're
talking about reactions then here also
dr by dt is some most of the time
directly proportional to the rate the
rate of reaction is that will be
reaction in Newton's law of cooling we
extensively use this guy how because
Newton's law of cooling says that rate
of change of temperature the cavity's
temperature with respect time the small
DS time is nothing but minus k e- t
enviro t environ vironment enricher t is
the current temperature so if you see
this T is constant so if you see what
happens here is this guy is nothing but
this guy and this guy if you compare so
I'm having T here I am having DT by T
here correct so temperature and date of
trim change of temperature so here also
if you see T and derivative of T and
both are in a relation so this is also
example of definite
inflation if you talk about the atomic
decay the decay which you have so here
also if you see let's suppose ever the
other D is decay and D D by DT is here
also directly proportional to the D so
rate of decay szostak juda decay in case
of the circuits which we use in physics
you have this L inductor for inductor
the e which you have will be L di
ability for this particular inductor
circuit actually if you write the
equation my equation will be e is equal
to this will become L di by DT and plus
RI correct so here if you see for I I
have this variable here and this is only
one derivative of Y so I have an
equation where I have I and derivative
of I work so this is also one example of
study of differential equation so the
concept is pretty clear now any equation
which has any function and derivative or
the function also write with respect to
any any variable X here actually so if
you have a relationship between these
two in one function then that equation
is a differential equation or instead of
DF x by DF by DX I can take second
derivative also so any derivative squash
auto second-order you take in the
function and the other function you have
a relation that guy is differential
equation and the practical application
was the such kind of equation is in the
field of science mostly in the field of
science only population growth and stuff
like let's recap relate some notations
which we'll be using in this chapter
so if I have dy by DX something right so
where I have to find derivative of Y
this 2x I'll say this guy's nothing but
y dash D 2 y by DX 2 I'll write this guy
as y double dash similarly V cubed Y by
D X cube all right this guy's voice but
for Annette differentiation I won't
write
I got by with this for this i'll use d
NY by d this is the notation we'll be
using with d-y-by-dx I will use Y - B -
y by DX 2 I'll use y double dash B cube
Y by DX I use Y triple dash for any
other order of derivative more than 3
I'll use DN y by D X to the party
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