Differential Equation Part-5 ( Equation Reducible to Exact eqn.-I ), Electrical Engineering, GATE Video Lecture

hello guys this gate lecture of
engineering mathematics on topic
equation the disabled to exact equation
is brought you by lectures for free calm
hey guys this is a part 4 of
differential equation series if you
haven't seen part 1 to part 3rd then I
will recommend you to watch that lecture
also okay read so let's start and see
what are the topics that we will cover
in this lecture basically there are four
rules to convert any standard equation
of which is not exact to the exact
differential equation so in this lecture
we will see rule 1 and the rule 2 in the
simple questions related to rule 1 and
rule 2 rule 3 and rule for we will see
in our next lectures okay so let's start
the lecture equation reducible to exact
equations as we have seen earlier
what is exact equation executive what is
exact differential equation and the what
are the sufficient and necessary
condition for take that differential
equation as we have seen earlier let me
let me again let me again divide that
context any given equation which is in
the form of M X Y DX plus and X Y dy is
equal to 0 it will be exact differential
equation if del M upon del Y is not it
or it is equal to Del n upon del X if
this condition is satisfied then any
given equation which is in the form of
this will be
exact differentiation Italy exact
differential equation
and then we can find the solution of
exact differential equation either by
using this formula integration of MDX
keeping y equal to constant +
integration of terms of n not containing
X 6 into dy is equal to C or by using
this formula integration of n dy keeping
X equal to constant + integration of
terms of M not containing X so if not
containing Y DX is equal to C okay so if
del M upon del Y is equal to Del n upon
del X then this equation will be exact
but if the given equation is in this
form M X M DX plus n dy and it is non
non adept this dilemma 1 del Y is not
equal to Del n upon del X then then also
then also we can multiply the equation
with some vector a and it will convert
put to the exact differential equation
then we can find the solution by using
this equation or this equation so to
convert any non exact differential
equation to differential equation we
will multiply the equation with 1 factor
known as integrating factor known as
integrating factor okay
you you are familiar with distance
integrating factor
okay so in different cases integrating
factor will be different okay so to find
out the value of integrating factor
there are four rules there are four rule
and they depending on that rule we will
get the integrating factor and the B
will multiply integrating factor to the
power equation then our equation will be
converted to the exact differential
equation and then we can apply either
equation one or equation two to get the
solution of the equation okay so let's
first of all see the rule number one if
the given equation MX 5 DX plus n XY dy
is equal to zero is not exact but it is
what it is homogeneous then integrating
factor equals is equal to then our
integrating factor will be 1 upon MX
plus NY okay this condition will be
there then our integrating factor will
be 1 upon MX plus NY but under the
condition but under the condition MX +
NY this should not be equal to 0 ok if
this will be 0 then our integrating
factor will be infinite and if we will
multiply infinite to our equation then
our holy person will be infinite and we
will not get any solution
so this integrating factor is only wear
it when MX + NY is not equal to 0 ok
here one term ax came homogeneous ok
let's see what is no more genetic better
then only we can apply the rule number
one so let's see what is homogeneous
equation a function having the same
number of powers in every term you can
say it's like a function having same
number of power in every term in that
medical forms you can write like if any
function of KX and KY then if we can
write this function like a K to the
power n function of X Y then we will
call function f KX KY is the homogeneous
function of order n then we will call it
a homogeneous function of order n ok so
if you are not getting it then let's see
one example then you will understand
what will be the homogeneous equation
let us the example it's a square let's
suppose something of root of 8 XY is the
X square minus 3 XY plus y square okay
so homogeneous function is a function
having seen same number of power in
every term so here you can see here X to
the power X power is 2 in this term X
power is 1 Y power is 1 yes total power
is 2 here also total power is 2 so this
function FX Y is done homogeneous
homogeneous function of order
- okay similarly this we can learn by
observing the equation we can say that
it is homogeneous function or not okay
so let's see the rule one again if our
equation M DX plus n dy is equal to zero
is not exact but it is homogeneous then
our integrating factor will be this 1
upon MX plus NY but MX + n well should
not be equal to 0 okay so if our
equation will be exact differential
equation del n then we don't need to
multiply with any integrating factor so
all you can you can say integrating
factor will be 1 ok so you can say 1 and
this is like MX plus NY ok so here you
will get condition if the function will
be homogeneous then MX + NY is equal to
1 if function will be homogeneous sorry
if function will be exact differential
equation if equation will be showing
equation will be exact differential
equation exact differential equation
then MX + n Y will equal to 1 okay
MX + n value is equal to 1 the condition
will be valid now let's see one sample
person related to the truth one let's
suppose consider one question find the
solution of let's suppose we need one
question lady example the x square y -2
X y square DX
plus three x square y minus X cubed dy
is equal to this let's suppose this is
your question okay so by observing the
question you can find out that this
equation is in the form of MDX or better
you can write it like m XY function of x
squared DX plus n function of X Y dy is
equal to 0 okay so by observing this
that this equation is a form of M DX /
10 divine you can you can get idea that
this is a problem related to exact
differentiation differential equation
okay exact differential equation but if
you will if you will try to find out del
M upon del Y and del n upon del X then
you will see that this will not be equal
okay if this is not equal it means this
equation will not be exact differential
equation okay so if it is not exact
differential equation then we will try
to find out integrating factor okay so
to see you don't need to calculate even
this del M upon del Y or n del n upon
del X you do not need to calculate this
term also to know whether this function
it whether whether this equation is
exact differential equation or not
okay how you can observe m and n okay if
you will observe this is the M and this
is n maybe we can write it and this is n
if you will observe M then you can see
this is a homogeneous function of order
3 similarly n n is the homogeneous
function of order 3
homogeneous because here if football two
white-power one is total power at three
here it's to the power 1 by 2 power to
total power is 3 means this function is
the homogeneous function of order 3 and
this function is also you can be X to
the power 2 where to the power 1 X to
the power 3 this is the homogeneous
function of order 3 ok if m and n are
the homogeneous function of order 3
taken so you do not need to find out del
F upon del Y and del n upon del F to see
whether this is exact or not only you
need to do is calculate MX + N word if
this will be 1 MX + NY
if this is equal to 1 then this equation
will be exact differential equation and
if this is not one means this is not
exact differential equation we have to
multiply this equation with integrating
vector ok
so let's calculate MX + N 1 here M is
this and n is this
so MX means X into X square Y minus 2 X
Y square plus NY
so Y into n 3 X square Y minus X cube so
you will get here X cube y -2 X square Y
square plus 3 X square Y square minus X
cubed 1 so did a qy in digits were
excavating a little here - 2 X square Y
square plus 3 X square Y square you will
get here texas square Y square so as you
can see this is not equal to 1 this is
some quantity this is not equal to 1
means this equation is not exact
ok we have to multiply it
integrating factors so as we have seen
in rule number one when these m and n
are the modes in education then our
integrating factor will be 1 upon MX
plus NY ok we have already calculated
this tenets upon n1 here here this is
hemant upon N 1 we have already
calculated this this is XA square Y
square so our integrating factor will be
1 upon 8 d square square so this will be
our integrating sector ok so now we got
the integrating factor so to convert the
given equation into exact differential
equation let's multiply the equation
with integrating factor ok so let's
multiply the given equation this into
this finding it again so when we will
multiply integrating factor in this
equation in depth and side and right
hand side both then we will get 1 upon
a2 square Y square into X cubed y -2 6 y
square DX plus 1 upon X square y square
3 X square Y minus X cube dy is equal to
0 into you can say 1 upon X square Y
square so you will get 0 obviously here
when the now let's simplify it then you
will get the 1 upon y -2 by X DX plus K
upon y minus X upon Y square dy is equal
to 0
now you got this equation this equation
will be now exact differential equation
you can check it by using the Balaam
upon del Y is equal to Del n upon del X
you can check it that this equation is
different clear liquid exact
differential equation or not by using
this condition by using this experience
sufficient condition let's take it let's
calculate del M upon del Y so this is a
M so differentiate it with respect to Y
so you will get minus 1 upon Y square
differentiation of 1 upon Y is minus 1
upon Y square okay a differentiation of
-2 by X will be 0 because we are
differentiating with respect to Y so
this term our differentiation will be 0
now differentiate M with respect to X
this is n this is your n in this video
and visited n so differentiate this with
respect to X so first term will be 0
because it contains only why this term
will be 0 and second term will be minus
1 upon y square it is the common and
differentiation of X will be 1 so you
will get this term minus 1 upon Y square
means you are getting del M upon del Y
is equal to Del n upon del X ok so this
two are equal leaves now our equation
given equation is converted into
differential equation so it is converted
into exact differential equation okay
now we know the we can find out solution
of exact differential equation by using
any of this formula formula 1 or formula
so let's suppose I am applying the
formula 1 to solve the to talk to call
this person apply this formula means
integration of M DX keeping Y constant
plus 2
of em not containing X B Y okay I am
applying this formula in our question
okay let's apply integration of M here M
is this one the integration of M means 1
upon Y minus 2 by X DX plus integration
of terms of n not containing X so here
mm is here yes you wonder and it is n
return but determine contains X so we
will not write this term we will write
only 3 upon Y this step so you will get
here 3 upon Y dy okay is equal to
constant see here we have to keep Y is
equal to constant here we have tilted y
equal to constant now if integrate this
if we will integrate this then we will
get integration of 1 upon Y DX we are
keeping one upon wildfires constant so
you will get 1 over Y into X minus 1
upon 2 minus 2 1 upon X integration of 1
oh connected log X plus 3 log Y is equal
to C okay so now we can again simplify
simplify this equation by like this X by
Y minus minus 2 log X we can write it as
log X square plus 3 log Y we can write
that as plus log Y cube is equal to C
now we can again more we can more
simplify it by writing X is log Y cube
upon X square is equal to C minus X upon
Y transpose X upon Y invited bit and
we can write like Slavic log Y cube
minus log square we can write like this
so now you got the simplified answer
here is one this is one sort of solution
of your given differential equation okay
now let's see the rule to whatever rule
do you think if you're given equipment
which is in this one M X Y DX plus na to
aid us in payment another function of x
and y function of x and y is equal to
zero is not exact but it is in this form
F XY v plus G X Y X dy is equal to zero
okay then our integrating factor will be
then our integrating factor will be 1
upon M ay minus NY under the condition
MX minus NY it should not be equal to 0
ok you can understand if M X minus NY
will be 0 then you will get infinite and
if you will multiply in fact in your
equation then you will get infinite and
you will not get the solution ok so this
can this rule is applicable only when M
X minus NY should not be equal to 0 okay
and your given equation will be in this
form observers this equation carefully
here it is given it is a F X Y it is a
function of X Y it is an odd function of
x and y it is not like this a function
of x and y it is not like this it is
like if X Y means it's a function of X Y
not X & Y okay function of X Y into y
into here it is a function of X Y into X
divide
you see the example like this then you
will get more idea okay so in this case
you do not need to find the we do not we
do not need to check the different exact
differential equation by this like del n
upon del Y is equal to very and upon del
X Irma you can take like this also MX
minus NY is equal to 1 if you're the
given equation is in this form and MX
minus NY equal to 1 then your equation
will be exact differential equation ok
and we can understand if this it will be
1 then your integrating factor will be 1
upon 1 min 1 means you are multiplying 1
with the equation means your equation
remains same and it will definitely it
will be also the different exact
differential equation ok so in this case
you can directly verify that the given
equation is differential equation or not
by using this MX minus NY is equal to 1
okay now let's see the sample question
related to the tool to now let's write
the one question X Y fenders all son of
XY plus 2x square Y square into y DX
plus XY minus XY minus X square Y Square
and - it is / - accessorize that into X
dy is equal to 0 ok so as you will see
this equation then you can observe that
this equation is like is like M you can
say X Y DX
plus n XY dy is equal to do it so like
this when Z to the function of X Y you
can Y this is also the function of x and
y so this equation is like this MX where
DX plus n x squared DX equal to zero so
here you get the idea that this equation
is like the is the type of exact
differential equation
ok so rather if you will observe them
more closely then you can see m and n is
like this so how do you use if you love
so this is a XY here x and y square XY
square here it is XY XY that means these
two are the function of XY you can write
like this if it its function of XY x y
DX plus it is also the fun another
function of XY x dy is equal to 0 so you
can also see it is not a homogeneous
equation it is not homogeneous equation
because here the X power is 1 Y power is
1 total power is 2 here takes for it to
Y power to total power is 4 so it is not
homogeneous equation means we cannot
apply the rule 1 in this question okay
and it is like this so we can apply rule
2 and the check for in the check for
exact differential equation by using
this condition MX minus M Y is equal to
1 or not just take this one MX minus NY
is equal to 1 or not if it will be 1
Delta equation will be exact
differential equation if it is not 1
then this equation is not exact
differential equation and we have to
multiply it with integrating factor to
make it differential equation so now let
us find out
MX
and why and what the game here m is this
this term AZ of em okay this video to M
so M X you multiply X here so let's
write like this XY + 2 X square Y
squared Y into X plus 2 must be
unequally minus NY n is here X Y minus X
square Y square into X into y little
pinched by it if it where y square plus
2 X cubed Y cubed minus X square Y
square minus X cubed Y cube so if you
simplify it then this term and this term
will can you cancel we can sit in this
term minus minus plus this term will
dead you will get like a 3 X cubed Y
cube okay now here we can see MX minus
NY it is not equal to 1 means the given
equation is not exact
they cannot exact differential equation
so we have to come we have to convert it
into exact differential equation by
multiplying it with integrating factor
okay
in this case what will be our
integrating factor 1 upon M X minus N 1
we have already find the value of value
of M X minus n by and that is your 3 x
qy y cube so our integrating factor will
give 1 upon 3 X cube and Y cube this
will be our integrating factor so now
let's multiply this integrating factor
in our equation and both side left
right inside so we will get we will get
1 upon 1 upon 3 X cubed Y cube then X Y
minus so plus 2 X square Y square into y
DX plus plus half but 1 upon X 3 X cubed
Y cube then X Y minus 6 is minus X
square Y square dy is equal to 0 into 1
upon 3 X cube y q so this 3 this 3 will
cancel each other and the this will term
will be 0 multiplied by anything will be
0 so you will get you will get
simplified equipment like this 1 upon X
square y plus 2 by X DX plus 1 upon X Y
square minus 1 upon y dy is equal to 0
now we got the equation now we can you
can verify this equation will be exact
differential equation now this term is
your M this term is e of n now you can
verify by in this del F 1 del Y is equal
to Del n upon del X okay you can verify
but don't need to verify it will be
exact because we have already multiplied
with integrating factor you do not need
to again verify here ok you can directly
go further and then you can solve
directly focus the letter I am suppose I
am sorry by our first term equation like
integration of M
DX keeping y equal to constant plus
integration of terms of n not containing
not containing X dy is equal to C okay
so here M is this so you can write like
a 1 upon X square Y minus plus 2 by X DX
plus integration of terms of n not
containing X so as we can see here this
term contains the X so we will not write
this term we will only write this term
because this term does not contain X so
1 minus 1 upon y dy is equal to C so you
can simplify it here you will get u you
are treating Y as constant so you can
keep it out and when upon Y integration
of 1 upon X square will be minus 1 upon
it plus plus 2 by if you can 1 by X
integration is log X so 2 log 8 minus
integration of 1 upon 1 upon Y will be
log Y is equal to C okay so you got the
equation like this this is your solution
you can again simplify it like you can
provide it like a log X square upon Y is
equal to C plus 1 upon X Y like this so
this is your answer okay
so guys these are the some sample
questions if you want to if you want to
see the more salt questions so you can
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