Class 12 Exam > Class 12 Videos > Example : Create differential equation-1 (Part - 6)- Differential Equation, Maths, Class 12
Example : Create differential equation-1 (Part - 6)- Differential Equation, Maths, Class 12 Video Lecture
FAQs on Example : Create differential equation-1 (Part - 6)- Differential Equation, Maths, Class 12 Video Lecture
| 1. What is a differential equation? |  |
Ans. A differential equation is a mathematical equation that relates an unknown function to its derivatives. It involves the derivatives of the unknown function and represents the relationship between the function and its derivatives.
| 2. What is the importance of differential equations in mathematics? | |
Ans. Differential equations are of great importance in mathematics as they are used to model various real-world phenomena. They provide a way to describe the rate of change of a function and are extensively used in physics, engineering, economics, and other scientific fields.
| 3. Can you provide an example of a differential equation? | |
Ans. Sure. A commonly known example of a differential equation is the simple harmonic motion equation: d^2x/dt^2 + kx = 0, where x represents the position of an object, t represents time, and k is a constant.
| 4. What are the types of differential equations? | |
Ans. Differential equations can be classified into various types based on their order and linearity. Some common types include ordinary differential equations (ODEs), partial differential equations (PDEs), linear differential equations, nonlinear differential equations, and homogeneous differential equations.
| 5. How are differential equations solved? | |
Ans. The solution of a differential equation depends on its type and order. Various methods are used to solve differential equations, such as separation of variables, integrating factors, substitution, power series method, Laplace transforms, and numerical methods like Euler's method or Runge-Kutta method. The appropriate method is chosen based on the characteristics of the given differential equation.
Text Transcript from Video
hello friends this video on differential
equation part 6 is brought to you by
example your palm no more fear from exam
before watching this video please make
sure that you have watched part one to
part 5 take home example y is equal to e
to the power 2 X 8 las be a similar
thing here they are two variables a and
B differentiate twice the
differentiation why this becomes e to
the power 2 X or its expand this because
it has X BX conference is e into these
power 2 X plus B into X e to the power 2
X so a tweety power two ways to live and
a to everybody makes into two correct
that's what you will get if you
differentiate this guy plus all the
challenge is this guy you have to apply
the product rule here so this becomes
differentiate X that becomes 1 into e to
the power 2 X plus you keep X constant
differentially Liberty of X that becomes
equal about 2x correct so that's what
you get so this guy is 1 and if you want
to solve this further or you want to
keep this e to the power 2 X and in
divide 3x in one shot so what you get
here is e to the power 2 X is one place
here I'll put 1 this a plus BX class
this guy comes with the power X to the
power 2 X do same thing is rearranges
now this is why - now I have to find y
double dot also correct so let's find by
double dash or if you want to make your
life simple you observe that this whole
guy is y itself so you need not
differentiate once again
what I get you'll differentiate but it
will be simple for you nobody comes to y
plus B into these graduates so now if
you differentiate double differentiate
that you ever do any way that becomes 2y
that becomes 2y - plus 2 into B to the
power 2 X this is my final solution
actually the only thing I do is I
replace this guy B because as I told
earlier also I had this vision known as
oh my God why - I got my double - I told
this my final equation I have to find a
and B using these two equations so here
also have to find using am using these
two equations so let's see if I can find
B I can find B actually because if you
see in this equation y is equal to b is
equal to from this equation if you see b
into the part by 2x is nothing but y
dash minus 2y so i'll replace this guy
here so this becomes 2 y - plus 2 into y
dash minus 2 y under the matt hamill
equation if you rearrange top only this
becomes y double dash minus 2y and 2y
minus 4y - and minus 4y they come take
this side it becomes 4y and that is my
answer correct that's how it is
now our different question is easy
question only we have why we found why -
we found y double dash and then using
these two equations we found the value
of B right and then place it here and
this is my final equation well take home
example so here also vs see we have two
variables a and B so that means you
define y dash and y double dash that's
what she owes so let's do that first and
then we'll see how we can find the final
equation y dash is nothing but e to the
power thanks
a cos x will become a sign X into a
minus sine X here minus sine X so if you
see this is how it is at the new oh this
is this form e to the part e outside e
to the power x cos x plus b into e to
the power x silence so now since there
are two variables in terms of X you have
to apply product rule so first we don't
this fashion let me write in this
fashion to be more specific a is
constant first I will take this constant
differentiate this guy so this becomes e
to the power x cos x becomes minus sine
X then I had to take this guy constant
and differentiate this game so this
becomes e to the power x definition of
this constants is cos is awesome
similarly for B I will take this guy
constant differentiate this game e to
the power X collection of sine X is cos
x class now I differentiate this guy
into power X definitely e to power X and
sine X I'll take all so that's how it is
so if you actually arrange this e to the
power X e to the power all are about to
the power e to the power X only this is
a cos x
right and then you have B sine X here
you have minus a sine X and B kha'zix so
that is rearranges by taking cos X and
sine X part common so my y dash is
nothing but e to the power X is a common
for all so I think this guy outside
right for cos x I have this guy and this
day and both are plus so that becomes a
plus B causes correct for sine X or sine
X I have this guy and this guy that is
viene where my B is positive n is
negative right so this becomes plus B
minus a this is what I get is y -
correct now I have to find y double dash
so for y double dash you see it's almost
same as this here is a plus a and B
here's a plus B a B minus a so let's
just find y double dash first y double
dash will be in this case you have a
plus B common he to the power x cos x
cos you differentiate e to the power x
that becomes e to the power x cos x is
constant plus t Dvorak's comma constant
and cossacks you differentiate it
becomes minus sine x right
and this is again plus B - is constant
then e to power X sine X we have now you
take e to power X differential e to
power X sine X constant plus
differential you do power X you get e to
the power X only and sine is
differentiated kha'zix that is what you
get that is y double dash correct so
here also now we have so many cos x sine
X so let's again merge that those values
e to the power x common context what we
have known so let's see cos x in
you get a plus B a plus B and this is
plus B minus a plus B right
the whole thing is into cos x plus y
over sine X part you have B minus a
minus a and then you have minus of a
plus B minus a this is what you get
correct
see if you see this this guy a and he
cancel you get to be here and this in
this case b and b he cancels right so
what you get here is e to the power X
into cos x into to a right to a - you
get sine X into so here you will get
here you get to B to be here you get to
correct so or we can say that you can
take two also common so you get is equal
to 2 into e to the power X into B cos x
minus e sine X and that is nothing but
my Y Double Dash correct or to make
simple life my simple I'll just make
this by please well do this by to this
is gone okay so I have three equations y
is equal to do this value
my first equation y dash is equal to
this value my second equation and Y
Double Dash is equal to this one my
third equation so now from these three
equation I have to form it in such a way
that a and B is gone correct
so let's see what will happen if you see
my Y is of this form e to the power x
cos x beat the power sign x and y dash
by to use of this form e to the power X
because X minus a sine X
you add these two guys you may get this
guy by - let's do that
y + y - pi - double - by 2 is nothing
but e to the power X because X this guy
- a sine x + y why is this guy e to the
power X a cos x plus B sine X so you
take sine X cross X for common so this
becomes cos x becomes a plus B right it
lets me cross X it must be also X plus B
minus a sine X so if you observe this
guy this guy is nothing but equal to
this guy why - so this is nothing but by
- you see you disregard X a plus B cos x
in the bar X a plus B cos x e to power X
B - annex e to the power X penis - so
what do you observe that from my
observation upon that y plus y double
dash by 2 is y - and that is my vision
correct so you want to solve this
further you get y double dash minus 2 y
- minus sorry plus 2 y that's nothing
great also in this question the only
thing is we have to find y Y - y double
dash so we have three equations we can
just or we could have done this way also
we could have we have Y we have Y - 2 -
equation from this we could have find
the value of a and again in this
equation my this final equation but that
was a little difficult for me right
because finding a and B is a difficult
you know this guy system of a plus B
form and this guy is a form little
difficult for me
so what I did was I just
these equations and arranged in such a
way that I got that thank you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more
thanks and Sabine
equation part 6 is brought to you by
example your palm no more fear from exam
before watching this video please make
sure that you have watched part one to
part 5 take home example y is equal to e
to the power 2 X 8 las be a similar
thing here they are two variables a and
B differentiate twice the
differentiation why this becomes e to
the power 2 X or its expand this because
it has X BX conference is e into these
power 2 X plus B into X e to the power 2
X so a tweety power two ways to live and
a to everybody makes into two correct
that's what you will get if you
differentiate this guy plus all the
challenge is this guy you have to apply
the product rule here so this becomes
differentiate X that becomes 1 into e to
the power 2 X plus you keep X constant
differentially Liberty of X that becomes
equal about 2x correct so that's what
you get so this guy is 1 and if you want
to solve this further or you want to
keep this e to the power 2 X and in
divide 3x in one shot so what you get
here is e to the power 2 X is one place
here I'll put 1 this a plus BX class
this guy comes with the power X to the
power 2 X do same thing is rearranges
now this is why - now I have to find y
double dot also correct so let's find by
double dash or if you want to make your
life simple you observe that this whole
guy is y itself so you need not
differentiate once again
what I get you'll differentiate but it
will be simple for you nobody comes to y
plus B into these graduates so now if
you differentiate double differentiate
that you ever do any way that becomes 2y
that becomes 2y - plus 2 into B to the
power 2 X this is my final solution
actually the only thing I do is I
replace this guy B because as I told
earlier also I had this vision known as
oh my God why - I got my double - I told
this my final equation I have to find a
and B using these two equations so here
also have to find using am using these
two equations so let's see if I can find
B I can find B actually because if you
see in this equation y is equal to b is
equal to from this equation if you see b
into the part by 2x is nothing but y
dash minus 2y so i'll replace this guy
here so this becomes 2 y - plus 2 into y
dash minus 2 y under the matt hamill
equation if you rearrange top only this
becomes y double dash minus 2y and 2y
minus 4y - and minus 4y they come take
this side it becomes 4y and that is my
answer correct that's how it is
now our different question is easy
question only we have why we found why -
we found y double dash and then using
these two equations we found the value
of B right and then place it here and
this is my final equation well take home
example so here also vs see we have two
variables a and B so that means you
define y dash and y double dash that's
what she owes so let's do that first and
then we'll see how we can find the final
equation y dash is nothing but e to the
power thanks
a cos x will become a sign X into a
minus sine X here minus sine X so if you
see this is how it is at the new oh this
is this form e to the part e outside e
to the power x cos x plus b into e to
the power x silence so now since there
are two variables in terms of X you have
to apply product rule so first we don't
this fashion let me write in this
fashion to be more specific a is
constant first I will take this constant
differentiate this guy so this becomes e
to the power x cos x becomes minus sine
X then I had to take this guy constant
and differentiate this game so this
becomes e to the power x definition of
this constants is cos is awesome
similarly for B I will take this guy
constant differentiate this game e to
the power X collection of sine X is cos
x class now I differentiate this guy
into power X definitely e to power X and
sine X I'll take all so that's how it is
so if you actually arrange this e to the
power X e to the power all are about to
the power e to the power X only this is
a cos x
right and then you have B sine X here
you have minus a sine X and B kha'zix so
that is rearranges by taking cos X and
sine X part common so my y dash is
nothing but e to the power X is a common
for all so I think this guy outside
right for cos x I have this guy and this
day and both are plus so that becomes a
plus B causes correct for sine X or sine
X I have this guy and this guy that is
viene where my B is positive n is
negative right so this becomes plus B
minus a this is what I get is y -
correct now I have to find y double dash
so for y double dash you see it's almost
same as this here is a plus a and B
here's a plus B a B minus a so let's
just find y double dash first y double
dash will be in this case you have a
plus B common he to the power x cos x
cos you differentiate e to the power x
that becomes e to the power x cos x is
constant plus t Dvorak's comma constant
and cossacks you differentiate it
becomes minus sine x right
and this is again plus B - is constant
then e to power X sine X we have now you
take e to power X differential e to
power X sine X constant plus
differential you do power X you get e to
the power X only and sine is
differentiated kha'zix that is what you
get that is y double dash correct so
here also now we have so many cos x sine
X so let's again merge that those values
e to the power x common context what we
have known so let's see cos x in
you get a plus B a plus B and this is
plus B minus a plus B right
the whole thing is into cos x plus y
over sine X part you have B minus a
minus a and then you have minus of a
plus B minus a this is what you get
correct
see if you see this this guy a and he
cancel you get to be here and this in
this case b and b he cancels right so
what you get here is e to the power X
into cos x into to a right to a - you
get sine X into so here you will get
here you get to B to be here you get to
correct so or we can say that you can
take two also common so you get is equal
to 2 into e to the power X into B cos x
minus e sine X and that is nothing but
my Y Double Dash correct or to make
simple life my simple I'll just make
this by please well do this by to this
is gone okay so I have three equations y
is equal to do this value
my first equation y dash is equal to
this value my second equation and Y
Double Dash is equal to this one my
third equation so now from these three
equation I have to form it in such a way
that a and B is gone correct
so let's see what will happen if you see
my Y is of this form e to the power x
cos x beat the power sign x and y dash
by to use of this form e to the power X
because X minus a sine X
you add these two guys you may get this
guy by - let's do that
y + y - pi - double - by 2 is nothing
but e to the power X because X this guy
- a sine x + y why is this guy e to the
power X a cos x plus B sine X so you
take sine X cross X for common so this
becomes cos x becomes a plus B right it
lets me cross X it must be also X plus B
minus a sine X so if you observe this
guy this guy is nothing but equal to
this guy why - so this is nothing but by
- you see you disregard X a plus B cos x
in the bar X a plus B cos x e to power X
B - annex e to the power X penis - so
what do you observe that from my
observation upon that y plus y double
dash by 2 is y - and that is my vision
correct so you want to solve this
further you get y double dash minus 2 y
- minus sorry plus 2 y that's nothing
great also in this question the only
thing is we have to find y Y - y double
dash so we have three equations we can
just or we could have done this way also
we could have we have Y we have Y - 2 -
equation from this we could have find
the value of a and again in this
equation my this final equation but that
was a little difficult for me right
because finding a and B is a difficult
you know this guy system of a plus B
form and this guy is a form little
difficult for me
so what I did was I just
these equations and arranged in such a
way that I got that thank you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more
thanks and Sabine
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