Separation of Variables // Differential Equations
Summary
TLDRThis video introduces the method of separation of variables, a fundamental technique for solving differential equations. It demonstrates how to isolate variables to simplify equations, such as exponential growth models, which are applicable in various real-life scenarios like pandemic growth or bank account interest. The process involves dividing by the function of y, integrating both sides with respect to t, and solving for y. The video also addresses the complexity of finding solutions, including singular solutions like y=0, which are not always evident through the chosen method, highlighting the importance of exploring various approaches in differential equations.
Takeaways
- đ The video covers the method of separation of variables, a major method for solving differential equations.
- đ A free and open-source textbook and an entire course on differential equations are available in the video description.
- đ The exponential growth equation is used as an example to introduce differential equations, showing how the rate of change is proportional to the value of y.
- đ The solution to the exponential growth equation, y = C * e^(kt), is verified by the method of separation of variables.
- â By dividing the differential equation and integrating both sides with respect to t, the solution can be derived.
- đ A change of variables is introduced, defining dy as (dy/dt) * dt, allowing the integration to be carried out separately in terms of y and t.
- đ The method of separation of variables is applied to a more complex example, demonstrating the process of integrating both sides after separating variables.
- 𧩠The video emphasizes that some solutions to differential equations may be implicit and difficult to express explicitly.
- đĄ Singular solutions, such as y = 0 in the example, can exist and need to be considered separately from the implicit solutions derived from the method.
- đ The video highlights the importance of initial conditions in determining the specific solution curve for a given differential equation.
Q & A
What is the method of separation of variables in the context of differential equations?
-The method of separation of variables is a technique used to solve differential equations by rearranging the equation so that all terms involving the dependent variable (y) are on one side and all terms involving the independent variable (t or x) are on the other. This allows for the integration of both sides with respect to their respective variables.
Why is the exponential growth equation a good example for demonstrating the method of separation of variables?
-The exponential growth equation is a good example because it has a simple form where the rate of change of y is proportional to the value of y itself. This makes it easy to illustrate how the method of separation of variables can be applied to derive the solution y = Ce^(kt), where C is a constant.
What does the 'C' in the exponential growth solution represent?
-The 'C' in the exponential growth solution represents the constant of integration, which accounts for the initial conditions of the problem and ensures that the solution is general enough to fit any specific case.
How does the method of separation of variables help in solving real-life phenomena such as pandemic growth or bank account interest?
-The method of separation of variables helps in solving these phenomena by providing a mathematical model that describes how quantities change over time when the rate of change is proportional to the current amount, such as the number of infected individuals in a pandemic or the balance in a bank account with compounding interest.
What is the significance of the term 'separable differential equation'?
-A separable differential equation is one where the dependent and independent variables can be separated on either side of the equation, allowing for the application of the method of separation of variables to find a solution.
What is the process of integrating both sides of a separable differential equation with respect to time?
-The process involves moving all terms involving the dependent variable to one side and all terms involving the independent variable to the other side. Then, integrate both sides with respect to time, which may involve recognizing the differential dy as dy/dt and integrating with respect to t.
What is the difference between an implicit solution and an explicit solution in differential equations?
-An implicit solution is an equation that defines the relationship between the variables but does not explicitly solve for one variable in terms of the other. An explicit solution, on the other hand, provides a direct formula for one variable as a function of the other.
Why might a differential equation have more than one solution?
-A differential equation might have more than one solution because different methods of solving or initial conditions can lead to different forms of solutions. Additionally, singular solutions, like y = 0, can exist independently of the general solution found through separation of variables.
What is the role of the constant 'k' in the exponential growth solution?
-The constant 'k' in the exponential growth solution represents the growth rate of the function y over time. It determines how quickly or slowly y increases as time progresses.
How does the method of separation of variables handle the integration of terms that are not easily integrable?
-If the terms are not easily integrable, the method may still proceed by separating the variables and integrating what can be integrated, leaving the rest in its integral form. The resulting equation, even if not fully integrable, can still provide valuable insights into the relationship between the variables.
What is the significance of the singular solution y = 0 in the context of the given script?
-The singular solution y = 0 is significant because it represents a special case where the derivative of y is zero, satisfying the differential equation independently of the general solution found through separation of variables. It highlights the importance of considering all possible solutions when solving differential equations.
Outlines
đ Introduction to Separation of Variables
This video introduces the method of separation of variables, which is used to solve differential equations. The video is part of a course on differential equations, with links to the playlist and a free textbook in the description. The exponential growth equation is revisited, which models real-life phenomena where the rate of change is proportional to the current amount, such as in pandemics or continuously compounding interest.
𧟠Applying Separation of Variables
The method of separation of variables is demonstrated. By dividing both sides by \(y\), integrating with respect to \(t\), and using a change of variables, the video shows how to derive the solution \(y = C e^{kt}\). This involves manipulating the equation to isolate \(y\) and then performing integration.
đ General Methodology
A more general approach to separation of variables is discussed, where a first-order differential equation can be separated into a function of \(y\) and a function of \(t\). By integrating both sides with respect to \(t\) and using the change of variables technique, the solution can be found.
đ Example: More Complex Differential Equation
A more complex example of a differential equation, \(y' = \frac{xy}{y^2 + 1}\), is solved using separation of variables. The equation is separated, integrated, and results in an implicit solution, \( \frac{y^2}{2} + \ln|y| = \frac{x^2}{2} + C \). The concept of a singular solution, \( y = 0 \), is introduced and its implications are discussed.
𧩠Singular Solutions and Implicit Equations
The video explores the idea of implicit equations and singular solutions in differential equations. Using the example \( y' = \frac{xy}{y^2 + 1} \), it is shown how the implicit solution varies with different values of \( C \) and how the singular solution \( y = 0 \) fits into the overall solution set.
đ„ Conclusion and Next Steps
The video concludes by encouraging viewers to leave questions in the comments and to check out the differential equations playlist linked in the description. The instructor teases more math topics to be covered in the next video.
Mindmap
Keywords
đĄSeparation of Variables
đĄDifferential Equation
đĄExponential Growth
đĄIntegral
đĄConstant of Integration
đĄImplicit Solution
đĄSingular Solution
đĄLogarithm
đĄChange of Variables
đĄInitial Condition
Highlights
Introduction to the method of separation of variables as a major method to solve differential equations.
Application of the exponential growth equation to real-life phenomena such as pandemics and continuously compounding interest.
Explanation of the method of separation of variables by dividing the equation to separate y and t terms.
Integration of both sides of the equation with respect to t to solve for y.
Introduction of a convenient fiction to simplify the integration process by defining dy and dt.
Derivation of the general solution y = Ce^(kt) through exponential manipulation.
Generalization of the separation of variables method for any first-order differential equation.
Process of integrating both sides of the equation after separating variables.
Example of a more complex differential equation to illustrate the method.
Identification and handling of implicit solutions in the context of separation of variables.
Recognition of singular solutions that may not be captured by the separation of variables method.
Discussion on how singular solutions like y = 0 fit into the broader solution set.
Graphical representation of solutions for different values of the integration constant.
Insight into how initial conditions determine specific solution curves.
Acknowledgment of the complexities and challenges in solving differential equations, and an invitation to further explore these topics in the course.
Transcripts
in this video we're going to learn about
the method of separation of
variables which is going to be our first
major method to be able to actually
solve a differential equation this video
is part of an entire course on
differential equations and the link to
that playlist
as well as the free and open source
textbook that accompanies it
is down in the description now when i
first introduced differential equations
we talked about the
exponential growth equation an equation
where
on the left hand side there was a rate
of change of y a derivative and on the
right hand side
it was proportional to the value of y
and
this exponential growth equation models
a whole lot of real life phenomena
any time where the rate of change is
proportional to the amount that you
actually have for example in a pandemic
when we're in the early days
and the growth rate is just proportional
to the number of people who are infected
or in a bank account where the growth
rate if it's
compounding continuously is proportional
to the amount that you have perhaps
five percent interest as time goes on
and so this differential equation comes
up in all sorts of places
but when we previously talked about it i
just gave you the solution i said the
solution
is y is some constant times e to the kt
and you could verify that yes it was a
solution by plugging it into the
left-hand side by taking the derivative
studying that equal to the right-hand
side
but where did this come from that's what
i'm going to answer in this video
so what we're going to do is this method
of separation of variables and the first
thing i'm going to do is i'm going to
take the y on the right hand side and
i'm just going to divide it out and put
on the other side
and the reason i'm doing this is that
now on the left hand side there's
everything in terms of y and the
derivative with respect to y
on the right hand side there's nothing
to do with y at all
now i'm going to take an integral of
both sides with respect to t
i'm allowed to do that i'm allowed to
add the same thing to both sides i'm
allowed to
multiply both sides by something that's
non-zero i'm
allowed to integrate both sides with
respect to t i can do the same thing to
both sides of an equation so
i can integrate both sides of an
equation with respect to t
and then i notice that i have on the
left this d
y d t d t and it's tempting to just
say that i can cancel the dt divided by
dt which is sort of a convenient fiction
because what we're actually going to do
is define something new called d y
d y will be defined to be d y d t
d t this is just a change of variables
so now on the left i have an integral
entirely in terms of y
and on the right and integral entirely
terms of t let's do those integrals on
the left
the integral of one over y is the
logarithm of absolute value y on the
right the integral of k
becomes kt and then i always have to
remember to add the plus c
my additive constant of integration
okay so we have a solution here and we
actually can solve it a little bit
better i'm going to take e to the power
of both sides
to get rid of the logarithm so on the
left
e to the logarithm of absolute value of
y that's just going to cancel to become
absolute value of y and then the same
thing on the right
we're pretty close to being down here
but one more manipulation
e to the kt plus c when you add up in
the exponents
that's just the same thing as
multiplying by another copy e to the c
so
e to the kt plus c is the same thing as
e to the kt
times e to the c and then i'm just going
to
re-label e to the c is something called
c tilde
basically my additive constant up in the
exponent has now become a multiplicative
constant so c
and c tilde are slightly different but
either way it's just some constant
i also have dropped the absolute value
signs this was just because
exponential is always positive and so
don't have to worry about when y is
negative
and so now we've gotten the same
solution that i've asserted to you
previously by this method of separation
of variables okay
now let's step back and do this a little
bit more generally because this example
was actually so simplistic it might miss
some of the complexities of this method
so generically what i want to talk about
is when you have a
first order differential equation so a
single derivative y prime
and that that can be written as the
product of some stuff entirely in terms
of t
some function of t and some stuff
entirely in terms of y
some function g of y if it can be
expressed as a portion that's a function
of t and a portion that's a function of
y those are
multiplied together then you can use
separation of variables and here's how
the method works we'll do exactly what
we did before
everything with respect to y has been
moved to the left now so on the left i
have
y and derivatives of y that's what's
appearing on the left
on the right i have things in terms of t
only
for now i've got a d y dt and i'm going
to leave that as one
object one derivative i'm not going to
try to separate it just yet i'll talk
about that a bit more in a moment
so on the left it's like y and the
derivative of y
okay now i have an equation i'm going to
integrate both sides
of that equation with respect to t so
the integral of
1 over g of y d y dt dt is equal to the
integral of
f of t dt i integrated respect to t on
both sides
i'll do the same trick as before i will
define
the differential d y to just be d y d d
d t
and so i get to replace that and now i
have an integer with respect to y
and then it goes back to t as a matter
of
preference what a lot of people actually
do is just separate out the dy and the
dt and move them to the opposite sides
of the equations at the beginning
and then just integrate respect to those
two variables that's fine that's a
really good shorthand to do
in fact that's what i do as a shorthand
but it is still good to know that what
we're sort of
properly doing is integrating both sides
respect to t and then doing a change of
variables
you want to separate out the dy and the
dt early on that's okay with me
either way i have an equation of two
intervals
i might be able to do these intervals i
might not be integration can sometimes
be hard but
if i can do those integrals i'm gonna
get some equation in terms of y
and t but no longer in terms of y prime
it's an equation of y and t
and that will be my solution to the
separable differential equation
okay so let's look at a slightly more
interesting example here
this is now y prime is equal to xy
divided
by y squared plus one our first question
is is it separable and yes it is
we have a portion which is just the x
and that is multiplied
by a portion which is some function just
of y the y divided by y squared plus one
it's a multiplication of these two
separated components
so let's apply the methodology first of
all i'm going to divide through so
dividing through by the
function of y gives me y squared plus 1
on the top divided by y
times d y and then on the right hand
side x dx here i have now gone and taken
the d y d x and sort of as a
convenient fiction just separated it
from both sides so it's all y's on the
left and all x is on the right
okay so now i integrate both sides which
is great
and now it's just a matter of doing
those integrals so on the left hand side
okay y squared
divided by y is just y it integrates out
to y squared divided by two
one over y integrates to logarithm of
the y x integrates to
x squared divided by two and then of
course i always have to do this i have
to add that plus c
now this solution to the differential
equation
is maybe not the nicest thing you've
ever seen because what it is
is an implicit solution i don't know how
to solve this
and make it y equal to a function of x
i haven't nicely solved it the same way
i did with exponential growth we said
y equals constant times e to the kt
here it's some equation that governs the
relationship between y
and x here by the way i changed all of
this to be an integral respect to x
sometimes our
independent variable is x or t we should
be able to do both
but either way i have this implicit
equation that defines them and sometimes
it's just
all you get then that's okay however
there's even
one more complication which is that
there is another solution
to this differential equation one that
is not written
in this format it's pretty simple it's
just called y equal to zero
we sometimes call this the singular
solution i mean the derivative of y
equal to zero is just zero that's the
left-hand side would be zero and
if i plug in y equal to zero on the
right i'd get zero equal to zero so this
is a solution but y equal to zero
is not in the form of this implicit
equation that i have
this is a big challenge with
differential equations sometimes you get
solutions
but then there are even more that are
not discovered by the methodology that
you've chosen and
this is going to be a big theme that we
have to think about and resolve and
come up with some theory to discuss as
we go on in our course
to get a little bit of a sense of what's
going on with this implicit equation
i've come in here and i've typed it in
and graphed it for the specific value of
c
equal to zero and what you can see if i
come here and change the value of c
is that it all looks like the same basic
type of shape
but depending on where you start
depending on an
initial condition it would tell you
which specific value of c
you're in unless what specific plot are
you going to be on
and then it's also sort of worth noting
what's going on with the singular
solution y equal to zero which has not
been plotted here but if i wanted to i
could come along and
add an extra equation for y equal to
zero
well if i was to make c be very very
negative as you can see the more
negative c gets
the closer to that singular solution it
becomes
this sort of makes sense if the c value
is a very large negative
how would you get a negative on the
left-hand side well you'd have to be
taking
the logarithm of a number that was very
very close to zero and then that's why
as you have c going very very far to the
negative
y is going to be approaching 0 in some
sort of limiting sense
but this y equal to 0 the singular
solution
nevertheless looks substantially
different
than the solution to this implicit
equation at any given point
and these types of relationships between
the solutions that you find in so-called
singular solutions are quite common
in differential equations all right i
hope you enjoyed this video if you have
any questions please leave them down in
the comments below
because they have an entire playlist in
differential equations the link to that
is in the description
and with that let's do some more math in
the next video
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