Separation of Variables // Differential Equations

Summary

This 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

• 📚 The video covers the method of separation of variables, a major method for solving differential equations.
• 🔗 A free and open-source textbook and an entire course on differential equations are available in the video description.
• 📈 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.
• 🔍 The solution to the exponential growth equation, y = C * e^(kt), is verified by the method of separation of variables.
• ➗ By dividing the differential equation and integrating both sides with respect to t, the solution can be derived.
• 🔄 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.
• 📐 The method of separation of variables is applied to a more complex example, demonstrating the process of integrating both sides after separating variables.
• 🧩 The video emphasizes that some solutions to differential equations may be implicit and difficult to express explicitly.
• 💡 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.
• 🌐 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

00:00

📚 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.

05:01

🧮 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.

10:01

🔍 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.

📊 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.

🧩 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.

🎥 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.

Keywords

💡Separation of Variables

Separation of Variables is a method used to solve differential equations. In the video, it is introduced as the first major method for solving differential equations by separating variables to opposite sides of the equation. This method allows for integration of each side with respect to its own variable, simplifying the process of finding the solution.

💡Differential Equation

A differential equation is an equation involving derivatives that represents how a quantity changes over time. The video focuses on solving differential equations, specifically those where the rate of change is proportional to the amount present, such as in exponential growth scenarios.

💡Exponential Growth

Exponential growth refers to the increase in quantity where the rate of growth is proportional to the current amount. In the video, this concept is exemplified by the exponential growth equation, which models real-life phenomena like population growth or continuously compounding interest in a bank account.

💡Integral

An integral is a mathematical operation used to find areas under curves, among other applications. The video demonstrates how integrals are used in the method of separation of variables to solve differential equations by integrating both sides of the equation with respect to time.

💡Constant of Integration

The constant of integration (denoted as C) is an arbitrary constant added to the result of an indefinite integral. The video explains its importance when integrating both sides of a differential equation and how it can be manipulated to solve for the general solution.

💡Implicit Solution

An implicit solution is a solution to a differential equation that defines the relationship between variables without explicitly solving for one variable in terms of another. The video provides an example where the solution to a separable differential equation results in an implicit equation, indicating the relationship between y and t.

💡Singular Solution

A singular solution is a specific solution to a differential equation that may not be represented by the general solution obtained through standard methods. In the video, y = 0 is identified as a singular solution to a particular differential equation, highlighting the complexities in finding all possible solutions.

💡Logarithm

A logarithm is the inverse operation of exponentiation, used to solve equations involving exponentials. The video illustrates how taking the logarithm of both sides of an equation helps in solving the differential equation by converting multiplicative relationships into additive ones.

💡Change of Variables

Change of variables is a method used to simplify an equation by redefining the variables involved. In the video, this technique is employed during the integration process to facilitate the separation of variables and the subsequent integration of each side.

💡Initial Condition

An initial condition specifies the value of the solution at a particular point, used to determine the specific constant of integration for a differential equation. The video discusses how initial conditions affect the value of the constant C, which in turn determines the specific solution plot.

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