Class 12 Exam > Class 12 Videos > General & particular solution - Differential Equations
General & particular solution - Differential Equations Video Lecture - Class 12
FAQs on General & particular solution - Differential Equations Video Lecture - Class 12
| 1. What is the difference between a general solution and a particular solution in differential equations? |  |
Ans. In differential equations, a general solution refers to a solution that includes all possible solutions to the equation. It contains arbitrary constants that can be assigned any value. On the other hand, a particular solution is a specific solution obtained by assigning specific values to the arbitrary constants in the general solution. It satisfies the given initial conditions or boundary conditions.
| 2. How can we find the general solution of a differential equation? | |
Ans. To find the general solution of a differential equation, we typically follow these steps:
1. Solve the differential equation by separating variables, using integrating factors, or applying other suitable methods.
2. Integrate both sides of the equation to find a general expression involving an arbitrary constant(s).
3. Simplify the equation if necessary and express the solution in its most general form, with arbitrary constants representing all possible solutions.
| 3. How do we determine a particular solution of a differential equation? | |
Ans. To determine a particular solution of a differential equation, we need additional information such as initial conditions or boundary conditions. By substituting these specific values into the general solution, we can solve for the arbitrary constants and obtain the particular solution that satisfies the given conditions. This particular solution is unique for a given set of initial or boundary conditions.
| 4. Can a particular solution also be a general solution? | |
Ans. No, a particular solution cannot be considered a general solution. A particular solution is specific to a particular set of initial or boundary conditions, meaning it satisfies those conditions exactly. In contrast, a general solution includes all possible solutions to the differential equation and contains arbitrary constants that can take on any value. These constants allow the general solution to represent an infinite family of solutions.
| 5. What is the purpose of finding both the general solution and particular solution in differential equations? | |
Ans. The general solution of a differential equation provides a comprehensive expression that encompasses all possible solutions. It allows us to understand the behavior and characteristics of the solutions in a more general sense. On the other hand, the particular solution is essential for specific applications where we need to find the solution that satisfies given initial or boundary conditions. It provides a unique solution tailored to those conditions, enabling us to solve real-world problems accurately.
Text Transcript from Video
hello friends this video on differential
equation part 3 is brought to you by
exam clear calm no more fear from exam
before watching this video please make
sure that you have watched part 1 and
part 2 now we have understood what is
degree in order let's find the general
and particular solution of a
differential equation so let's compare
the differential equations in the normal
equations we have so the normal equation
we know we know we are aware of this
kind of the efficient it's also an
equation X square minus 1 will f 0 then
we say X 1 - 1 or minus 1 right or we
say X plus 2 is equal to 0 you get X
equal to minus T these kind of equations
we know where you have one or two
solutions they always particular
solution because the maximum it all
depends on the power of this so if it
has 2 power to have at the max 2
solutions if it has one power at the
maximum earlier these kind of equations
we know and the value of the X is all
this numbers please note here I am
getting X as 1 or minus 1 right in this
case I'm going X Delta minus 2 so value
of x is obvious number but a
differential equation as I told is of
this this time is a tool dy by DX plus K
X K Y K X values in this kind of form
all right now in this case the value of
y I get is function please note in this
case the value of x you get is numbered
the logic it is number in this case I
get functions now so it is a function it
has to label solution it can be a
general solution or a particular
solution example this case I'll take if
I gain dy by DX is equal to minus sine X
or I get dy is equal to minus sine X DX
I just solve this the crude method
without even using the steps integrate
both side but I get Y is equal to cos x
plus multi so if you see this guy is a
general solution because my constant
value C is not known I can put the value
of cost as 1/2
four five ten this geyser general
solution but if I put the value of
constant example if I know the value of
this curve I know some y or x value I
can put this value of example y is equal
to zero when cost in the pipe I do
something of that sort but then I get
the value of C also and let's suppose C
came out to be eight then I write my
solution it has to be y is equal to cos
x plus n so in this case if you see I
got a particular solution because I know
this is the solution right only one
solution but in this case I am Telling Y
is going to cause it per C where C is
any variable so they can be multiple
solution with some value C right so such
solution is a general solution where you
have some variables defined and you say
okay this value can can have any value
and those you can have multiple
solutions that is the general solution
but when you have the actual value of
that then this is only one solution that
becomes a particular solution so so the
reason why we are slide was to tell you
two things one is the solution for
differential equation is not number its
function always second thing the
solution of two type either general or
so having said this let's take some
example to find solution of differential
equation by the few way
so in this case we're doing a reverse
fit since we don't know yet how to solve
the differential equation but we can
verify another particular solution as
part of the visit not so what I'm doing
is you to verify that this particular y
is equal to e to power X is my solution
which this function actually and this is
a particular solution right this fits
this differential equation over so if I
have my differential equation I have my
solution given I have to check whether
this particular solution is for this
differential equation or not for this I
need not know how to solve the
differential equation correct so we have
to verify that this given solution is
fitting to this corresponding
differential equation so differential
equation is y double dash
- why - is equal to you so I had to find
y double dash I have to find by - so let
us find y double dash if you already
find wide and also define y dash much
then you'll find my double dash so if my
y is equal to e to power X plus 1 my
dash will be e to the power x + 0 + y
double dash will be e to the power x + 0
itself correct why because my y is equal
to e to the power x + 1 you
differentiate this guy e to the power x
become in the power x itself 1 becomes 0
here also 0 becomes 0 only that's how I
get Y - and my double - so my
differential equation is this so let's
put the value of this game y double dash
is e to the power X plus 0 is 0 only and
y dash is also same so you see this guy
becomes 0 so this actually fits to this
solution then I can say that my this
solution is a solution for this
differential equation ok
so I did I do not get how to find the
solution perturbation equation but now
what I have done is given a solution of
the differential equation I verified
where that that solution is correct or
not
let's see comb example y is going to
cause X plus C this is if you see is a
general solution please note it is not
general solution is constant come here
and this is my differential equation I
have to verify if this solution is for
this equation
so this equation needs Y - let's find y
dash y dash becomes minus sine X C
becomes this is my violation so my
equation say is y dash plus - Villa 0
what is y X minus sine X correct plus
sine X and that is true actually so this
is true I can say that this particular
solution is true for this differential
equation let us take one more example
that will the fan
here so this is my solution here right
this is my solution and this is my
differential equation I have to prove
whether this solution is true for this
equation for this I have to find y -
that's Electrify
so let's find y dash y dash becomes to
differentiate X this becomes one into
sine X plus you take this guy common X
differentiate sine X this becomes
castle's so this becomes sine X plus X
forces this is my Y - I use the product
rule actually differentiate this guy
keep this guy constant keep this guy
constant differential risk so I got Y -
now you check whether this guy fits in
this so my equation is X Y - y - is this
guy sine x + 6 cos x is equal to y y is
again X sine X and putting everything in
terms of X plus takes into luto 6 square
minus y square Y square is nothing but X
square sine square X correct so I have
to prove LHS is equal to RHS correct so
this is my allegis and this is my
allergies so I will take other signal
expand more RHS this becomes X sine X
plus X into X into 1 minus sine square X
correct and 1 minus sine square has cos
square X this becomes X sine X plus X
into of X square or
and this becomes nothing word X 9 X plus
X square costs and if you see this guy
is exactly same as mine and the chess
you should take X common so this becomes
so if you see this guy is equal to LX is
it so I can prove that my LHS go to our
essence that means this is true correct
so this kind of questions are not
difficult you have the solution given
you have two equation given you just
have to put that solution in this
equation so I did that I put y is equal
to X sine x + y - I was to find - why -
from this equation only I got - build of
X sine X plus s cos x so I would void as
you can do - + S Halas right and I check
whether this was equal alexi lalas
let's take on my example y is equal to
root of 1 plus x square is my solution
for this differential equation y - 2 XY
by 1 plus X Y correct to find that I
first had to find y - also because this
implies y + bi - so let's find y dash y
dash is nothing but this is to the power
1 by 2 so this becomes 1 by 2 into 1
plus x squared if all minus 1 by 2 and X
square becomes students correct so this
becomes 2 and to cancel this becomes X
by 1 plus X now
correct and this is my alleged abusive
let's mind our HS takes Y by 1 plus X
square this is nothing but X Y is
nothing but food 1 plus X square by 1
plus X square if you cut this this
becomes X by 1 plus X square Y because
this guy's nothing but root 1 plus I
squared the party and this guy if you
see is nothing but my LHS because I have
proved that RHS is equal to X if R is
equal to a lecture that means this
particular function is a solution for
this
thank you visit exam viacom to watch
free education videos try free on my
debts get the best quality study
materials study from the best tutors and
mentors and much more thanks again
equation part 3 is brought to you by
exam clear calm no more fear from exam
before watching this video please make
sure that you have watched part 1 and
part 2 now we have understood what is
degree in order let's find the general
and particular solution of a
differential equation so let's compare
the differential equations in the normal
equations we have so the normal equation
we know we know we are aware of this
kind of the efficient it's also an
equation X square minus 1 will f 0 then
we say X 1 - 1 or minus 1 right or we
say X plus 2 is equal to 0 you get X
equal to minus T these kind of equations
we know where you have one or two
solutions they always particular
solution because the maximum it all
depends on the power of this so if it
has 2 power to have at the max 2
solutions if it has one power at the
maximum earlier these kind of equations
we know and the value of the X is all
this numbers please note here I am
getting X as 1 or minus 1 right in this
case I'm going X Delta minus 2 so value
of x is obvious number but a
differential equation as I told is of
this this time is a tool dy by DX plus K
X K Y K X values in this kind of form
all right now in this case the value of
y I get is function please note in this
case the value of x you get is numbered
the logic it is number in this case I
get functions now so it is a function it
has to label solution it can be a
general solution or a particular
solution example this case I'll take if
I gain dy by DX is equal to minus sine X
or I get dy is equal to minus sine X DX
I just solve this the crude method
without even using the steps integrate
both side but I get Y is equal to cos x
plus multi so if you see this guy is a
general solution because my constant
value C is not known I can put the value
of cost as 1/2
four five ten this geyser general
solution but if I put the value of
constant example if I know the value of
this curve I know some y or x value I
can put this value of example y is equal
to zero when cost in the pipe I do
something of that sort but then I get
the value of C also and let's suppose C
came out to be eight then I write my
solution it has to be y is equal to cos
x plus n so in this case if you see I
got a particular solution because I know
this is the solution right only one
solution but in this case I am Telling Y
is going to cause it per C where C is
any variable so they can be multiple
solution with some value C right so such
solution is a general solution where you
have some variables defined and you say
okay this value can can have any value
and those you can have multiple
solutions that is the general solution
but when you have the actual value of
that then this is only one solution that
becomes a particular solution so so the
reason why we are slide was to tell you
two things one is the solution for
differential equation is not number its
function always second thing the
solution of two type either general or
so having said this let's take some
example to find solution of differential
equation by the few way
so in this case we're doing a reverse
fit since we don't know yet how to solve
the differential equation but we can
verify another particular solution as
part of the visit not so what I'm doing
is you to verify that this particular y
is equal to e to power X is my solution
which this function actually and this is
a particular solution right this fits
this differential equation over so if I
have my differential equation I have my
solution given I have to check whether
this particular solution is for this
differential equation or not for this I
need not know how to solve the
differential equation correct so we have
to verify that this given solution is
fitting to this corresponding
differential equation so differential
equation is y double dash
- why - is equal to you so I had to find
y double dash I have to find by - so let
us find y double dash if you already
find wide and also define y dash much
then you'll find my double dash so if my
y is equal to e to power X plus 1 my
dash will be e to the power x + 0 + y
double dash will be e to the power x + 0
itself correct why because my y is equal
to e to the power x + 1 you
differentiate this guy e to the power x
become in the power x itself 1 becomes 0
here also 0 becomes 0 only that's how I
get Y - and my double - so my
differential equation is this so let's
put the value of this game y double dash
is e to the power X plus 0 is 0 only and
y dash is also same so you see this guy
becomes 0 so this actually fits to this
solution then I can say that my this
solution is a solution for this
differential equation ok
so I did I do not get how to find the
solution perturbation equation but now
what I have done is given a solution of
the differential equation I verified
where that that solution is correct or
not
let's see comb example y is going to
cause X plus C this is if you see is a
general solution please note it is not
general solution is constant come here
and this is my differential equation I
have to verify if this solution is for
this equation
so this equation needs Y - let's find y
dash y dash becomes minus sine X C
becomes this is my violation so my
equation say is y dash plus - Villa 0
what is y X minus sine X correct plus
sine X and that is true actually so this
is true I can say that this particular
solution is true for this differential
equation let us take one more example
that will the fan
here so this is my solution here right
this is my solution and this is my
differential equation I have to prove
whether this solution is true for this
equation for this I have to find y -
that's Electrify
so let's find y dash y dash becomes to
differentiate X this becomes one into
sine X plus you take this guy common X
differentiate sine X this becomes
castle's so this becomes sine X plus X
forces this is my Y - I use the product
rule actually differentiate this guy
keep this guy constant keep this guy
constant differential risk so I got Y -
now you check whether this guy fits in
this so my equation is X Y - y - is this
guy sine x + 6 cos x is equal to y y is
again X sine X and putting everything in
terms of X plus takes into luto 6 square
minus y square Y square is nothing but X
square sine square X correct so I have
to prove LHS is equal to RHS correct so
this is my allegis and this is my
allergies so I will take other signal
expand more RHS this becomes X sine X
plus X into X into 1 minus sine square X
correct and 1 minus sine square has cos
square X this becomes X sine X plus X
into of X square or
and this becomes nothing word X 9 X plus
X square costs and if you see this guy
is exactly same as mine and the chess
you should take X common so this becomes
so if you see this guy is equal to LX is
it so I can prove that my LHS go to our
essence that means this is true correct
so this kind of questions are not
difficult you have the solution given
you have two equation given you just
have to put that solution in this
equation so I did that I put y is equal
to X sine x + y - I was to find - why -
from this equation only I got - build of
X sine X plus s cos x so I would void as
you can do - + S Halas right and I check
whether this was equal alexi lalas
let's take on my example y is equal to
root of 1 plus x square is my solution
for this differential equation y - 2 XY
by 1 plus X Y correct to find that I
first had to find y - also because this
implies y + bi - so let's find y dash y
dash is nothing but this is to the power
1 by 2 so this becomes 1 by 2 into 1
plus x squared if all minus 1 by 2 and X
square becomes students correct so this
becomes 2 and to cancel this becomes X
by 1 plus X now
correct and this is my alleged abusive
let's mind our HS takes Y by 1 plus X
square this is nothing but X Y is
nothing but food 1 plus X square by 1
plus X square if you cut this this
becomes X by 1 plus X square Y because
this guy's nothing but root 1 plus I
squared the party and this guy if you
see is nothing but my LHS because I have
proved that RHS is equal to X if R is
equal to a lecture that means this
particular function is a solution for
this
thank you visit exam viacom to watch
free education videos try free on my
debts get the best quality study
materials study from the best tutors and
mentors and much more thanks again
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Dec 21, 2024 Last updated |
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