What is a DIFFERENTIAL EQUATION?? **Intro to my full ODE course**

Summary

This video introduces the concept of differential equations as a crucial tool for modeling the physical world, with a focus on their applications in various scenarios such as bank accounts, pandemics, and physics. The instructor outlines the basics of differential equations, demonstrates how to derive solutions, and emphasizes the importance of initial conditions in determining specific solutions. The video is part of a comprehensive playlist and open-source textbook designed to accompany a university course on the subject.

Takeaways

• 📚 Differential equations are powerful tools for modeling and understanding the physical world.
• 🎥 This video is the first in a series accompanying a university course on differential equations.
• 📘 A free and open-source textbook is available, corresponding to the video series.
• 🔢 A differential equation involves an equation with derivatives, often modeling how things change.
• 💰 Example: The rate of change of a bank account with a 3% interest rate can be modeled using a differential equation.
• 🦠 Exponential growth models are relevant for situations like bank accounts, pandemics, and bacterial growth.
• 📝 There is often an infinite family of solutions to a differential equation, dependent on initial conditions.
• ⚖️ Newton's second law can be used to derive differential equations for physical systems, like a ball thrown vertically.
• 📊 Solutions to differential equations can be found by integrating, with initial conditions determining specific solutions.
• 🔍 Future topics will cover when differential equations have solutions, the uniqueness of solutions, and methods to solve them.

Q & A

What is a differential equation?

A differential equation is an equation that includes derivatives, which is used to describe the rate at which a quantity changes in relation to other quantities, often time in the context of the script.

Why are differential equations important in modeling the physical world?

Differential equations are important because they allow us to describe and understand how different aspects of the physical world change over time, such as the movement of objects, the flow of fluids, and changes in electric and magnetic fields.

What is the prerequisite for the university course on differential equations mentioned in the script?

The prerequisite for the university course on differential equations is calculus, as it provides the foundational understanding of rates of change and derivatives, which are essential for studying differential equations.

What is an example of a differential equation that models a bank account with an interest rate?

An example given in the script is the differential equation dy/dt = 0.03 * y, where y represents the amount of money in the bank account at time t, and 0.03 is the continuous interest rate.

How does the solution e^(0.03t) relate to the bank account differential equation?

The function e^(0.03t) is a solution to the bank account differential equation because when you take its derivative, it results in the same function multiplied by the interest rate, satisfying the differential equation.

What does it mean when there are multiple solutions to a differential equation?

When there are multiple solutions to a differential equation, it means that there is a family of solutions that can be described by a constant, such as 'c' in the script, which can take on any value to produce a different solution.

What is an initial condition and why is it necessary in solving differential equations?

An initial condition is a specific value or condition at a given point in time, such as the amount of money in a bank account at time zero. It is necessary to find the particular solution that fits the specific situation being modeled.

How does the script illustrate the concept of exponential growth using differential equations?

The script illustrates exponential growth by showing that the rate of change of a quantity, such as money in a bank account or infections in a pandemic, is proportional to the quantity itself, leading to a solution that grows exponentially over time.

What is the second example of a differential equation presented in the script?

The second example is a differential equation derived from Newton's second law, modeling the motion of a ball thrown straight up into the air, where the only force acting on it is gravity.

How does the script explain the process of solving the second example of a differential equation?

The script explains the process by integrating the differential equation twice to find the position function of the ball over time, resulting in a quadratic equation that describes the ball's motion.

What are the two initial conditions needed to specify the solution of the second example of a differential equation?

The two initial conditions needed are the initial position (y naught) and the initial velocity (v naught) of the ball when it was thrown.

What is the significance of the number of derivatives in a differential equation in relation to the number of initial conditions required?

The number of derivatives in a differential equation determines the number of constants of integration, which in turn dictates the number of initial conditions needed to specify a unique solution.

What are some of the key questions that the future videos in the playlist will address?

The future videos will address questions such as when differential equations have solutions, the nature of these solutions (one or many), and the procedures for finding solutions to different types of differential equations.

Outlines

00:00

📚 Introduction to Differential Equations

The script introduces the concept of differential equations as a fundamental tool for modeling the physical world. It highlights the upcoming series of videos and an accompanying open-source textbook on differential equations, which are aimed at second-year university students with calculus as a prerequisite. The script explains that differential equations involve derivatives and are used to describe how things change, such as the growth of a bank account with a continuous interest rate. It also provides an example of a differential equation and its solution, emphasizing that often there are multiple solutions depending on the initial conditions.

05:02

🚀 Applying Differential Equations to Real-World Scenarios

This paragraph delves into the application of differential equations to model real-world phenomena, such as the growth of a bank account, the spread of a pandemic, and bacterial growth in a petri dish. It discusses the importance of initial conditions in determining the specific solution to a differential equation from the family of solutions. The script then presents a second example involving the motion of a ball thrown vertically, illustrating how differential equations can be derived from Newton's second law and solved to describe the ball's trajectory. The paragraph concludes by emphasizing the need for initial conditions to specify the constants in the solution.

10:04

🔍 Exploring the Theory of Differential Equations

The final paragraph of the script raises several questions about the nature and solutions of differential equations. It ponders the existence of solutions, the number of solutions, and the methods for finding them. The script acknowledges that while some solutions can be asserted or computed directly, there are many situations where specific procedures are needed to solve differential equations. It sets the stage for future videos that will explore these topics in depth, including the theory behind when solutions exist and how to determine them.

Keywords

💡Differential Equations

Differential equations are mathematical equations that involve derivatives, which are used to describe the rate of change of a function. In the context of the video, differential equations are presented as powerful tools for modeling and understanding the physical world. The video introduces the concept by explaining how these equations can represent various phenomena, such as the growth of a bank account with interest or the spread of a pandemic, where the rate of change is proportional to the current state.

💡Derivative

A derivative in calculus represents the rate at which a quantity changes with respect to another quantity. In the video, the derivative is used to model the rate of change of different systems, such as the amount of money in a bank account or the number of infections in a pandemic. The script uses the derivative to illustrate the foundational concept that leads to the formulation of differential equations.

💡Dependent Variable

In the context of functions and differential equations, the dependent variable is the variable that depends on another variable, often denoted as 'y' in the video. It represents the quantity that changes, such as the money in a bank account over time. The dependent variable is central to the formulation of differential equations, as it is the function whose rate of change is being modeled.

💡Exponential Growth

Exponential growth refers to a process where a quantity increases at a rate proportional to its current value. The video uses the example of a bank account with a 3% interest rate compounded continuously to illustrate exponential growth. This concept is modeled using a differential equation where the derivative of the bank account balance is proportional to the balance itself.

💡Initial Condition

An initial condition is a fixed value or condition that is given at the starting point of a problem. In the video, the concept is used to specify the exact state of a system at a certain time, such as the initial amount of money in a bank account or the initial number of infections in a pandemic. Initial conditions are crucial for determining the unique solution to a differential equation.

💡Solution

In the context of differential equations, a solution is a function that satisfies the equation. The video explains that differential equations can have multiple solutions, such as the exponential function e^(0.03t) for the bank account example. The script also discusses how solutions can be found by integrating the differential equation or by using other mathematical methods.

💡Continuous Compounding

Continuous compounding is a method of calculating interest where interest is added to the principal continuously over time, rather than at discrete intervals. The video uses this concept to explain how the interest on a bank account can be modeled with a differential equation, where the rate of change of the account balance is a constant percentage of the balance itself.

💡Second Derivative

A second derivative is the derivative of a derivative, representing the rate of change of the rate of change of a function. In the video, the second derivative is used in the context of Newton's second law to model the motion of a ball thrown straight up, where the acceleration due to gravity is the second derivative of the ball's position with respect to time.

💡Integration

Integration is the mathematical process of finding a function that, when differentiated, yields another given function. In the video, integration is used to solve the differential equation modeling the motion of a ball. By integrating the equation, the script demonstrates how to find the function that describes the ball's position over time.

💡Kinematic Equations

Kinematic equations describe the motion of an object under the influence of forces, such as gravity, without considering the forces themselves. The video refers to kinematic equations when discussing the motion of a ball thrown upwards, which is a classic example of using differential equations to model physical phenomena.

💡Constants of Integration

Constants of integration arise when integrating a differential equation. They represent the arbitrary constants that can be added to the integrated function without affecting the derivative. In the video, the script explains that the constants of integration, such as 'c' and 'd', are determined by initial conditions, which are necessary to specify the unique solution to the differential equation.

Highlights

Introduction to differential equations as powerful tools for modeling and understanding the physical world.

Announcement of a complete playlist and open source textbook accompanying a university course on differential equations.

Definition of a differential equation as an equation involving derivatives.

Explanation of how derivatives can be used to model rates of change in various real-world scenarios.

Illustration of a bank account model with an interest rate using a differential equation.

Introduction of the concept of exponential growth in the context of a bank account and its mathematical representation.

Demonstration of how to derive and verify a solution to a differential equation.

Discussion on the infinite family of solutions for differential equations and the role of constants.

Importance of initial conditions in determining specific solutions to differential equations.

Application of differential equations to model exponential growth in pandemics and bacterial populations.

Introduction of Newton's second law and its relation to differential equations in physics.

Derivation of a differential equation for a ball thrown vertically under the influence of gravity.

Solution of the vertical motion differential equation resulting in a quadratic function.

Explanation of how initial conditions affect the specific path of the ball's motion.

Connection between the number of derivatives in a differential equation and the number of initial conditions needed.

Discussion on the existence and multiplicity of solutions to differential equations.

Emphasis on the importance of learning procedures to solve various types of differential equations.

Invitation to engage with the video series and share resources with others studying differential equations.

Transcripts

welcome to differential equations

differential equations is one of the

most powerful tools that we have for

modeling and understanding the physical

world and in this video

i'm going to give you an introduction to

the big ideas of differential equations

now i'm particularly excited because

this video was just the first video in

an

entire playlist on differential

equations that accompanies the

university course that i am teaching on

differential equations it's a

pretty standard second year introduction

into differential equations that picks

calculus as a prerequisite

by the way in addition to this video

playlist there is also

a free and open source textbook that i'm

adapting that is corresponding to this

video series and you can check out the

links to both of those

down in the description so what is a

differential equation

when trying to model the world one of

the things that we notice is that things

change different objects move around

fluid

flows electric fields and magnetic

fields

they change stock prices go up and down

the world is changing and we have seen a

little bit of how to address how to get

our handle on a changing world in

calculus we've seen for example that if

you start with some function like

function y of t

then you can compute the rate of change

of that function or the derivative of

that function

y prime of t which was great and maybe

that would be helpful for modeling all

these changing things in the world

except

where did the original functions come

from as

in if you give me a function okay then i

could differentiate and get its rate of

change i could differentiate

twice if i wanted to and one common way

that they come from is called

differential equations and it's sort of

working this process

in reverse now the big idea is that

often when we're trying to

mathematically model a situation we know

more about

how something is changing than being

able to

exactly describe the state of the object

at every point in time

for example consider a bank account we

often are told

right up front what the rate of change

of a bank account is that will be our

interest

rate so consider this equation that

might model the scenario

the derivative of y is 0.03

times y y here is going to represent the

amount of money in our bank account at

time t and we typically in differential

equations do use

y for our dependent variable and what's

happening is that this is a bank account

with an interest rate of three percent

the fancy terminology is called three

percent compounded continuously

and what that means is whatever amount

of money that i have at a given point in

time

why the rate of change of increase

is 0.03 in other words 3 times that

amount of money

if you have more money than i do then

the rate of change

of your bank account is going to be

higher than mine because it's always

going to be the three percent of what

you

have and so this is a differential

equation a differential equation is just

an equation that has

derivatives in it so what can we do with

this well

i want to notice that this equation

actually has a solution

you can try it out if you accept that y

of t

is e to the 0.03 t it just works i mean

plug it into both sides

if you plug it into the left the

derivative of e to the 0.03 t

is well 0.03 times e to the 0.03 t

it's the same thing on both sides so

this

function e to the 0.03 t we say is a

solution to that differential equation

you might have asked how did i know that

that was it how are we going to come

up with solutions to differential

equations and we'll talk a lot about

that

more coming up in future videos right

now i just want to say

given this function that i've just

thrown at you well

yes indeed it is a solution to that

differential equation

and this kind of exponential growth well

it works for bank accounts but it also

works for things like a pandemic when we

were having

an exponential growth of a pandemic was

the same thing

the rate of change of infections was

some constant

times the number of people who already

had the infection

similarly the rate of change of some

bacteria growing in a petri dish

might be a constant multiple of the

population

in all these examples differential

equation was relevant because we could

say something about the rate of change

we could model it by making a claim

about the rate of change that the rate

of change

was proportional to the original values

y

now i can go even further than this

because well e to the 0.03 t is a

solution

but notice that 17 times e to the 0.03 t

is also a solution i can plug it into

both sides and i get that they are equal

and in fact the 17 was just a

placeholder it could have been anything

could also just be a

value of c and this gives us our first

lesson about differential equations

often there is not one solution to a

differential equation

there is an infinite family of solutions

for every value of c

we're going to get a different solution

so how can you find the values of c you

actually need

a little bit more information so i want

to redo the problem i want to stay with

the same solution that i have to that

differential equation

but i'm going to additionally oppose

something called an initial condition

an initial condition is you telling me

well

how much quantity was there at time t

equal to zero in this case i'm going to

say

i started with a thousand dollars at

time t equal to zero or there were

a thousand people infected at time t

equal to zero

depending on what it is that i'm trying

to model and so with an initial

condition if i plug t

equal to zero into that equation i just

get a thousand on the left as the

constant times e to the zero

e to the zero is one and that tells me

that c was a thousand

and so my y of t is a thousand times e

to the zero point zero three

t i can go over to desmos the

link to this is going to be down in the

description and plot what this looks

like

this is just exponential growth and so

as time goes

on my value of my bank account which

begins

at a thousand dollars is going to grow

and grow and grow at this three percent

compounded continuously

okay so that was one example of a

differential equation i want to leave

you with a second

and in this second example i'm going to

imagine i start with the ball i take the

ball

i throw it straight up and it falls down

to the ground

so base so the idea was it was just

moving vertically but i had a starting

spot

i had an initial velocity they gave it

and then it went up and back down until

it hit the ground

now if you remember any of your high

school physics what we have is that

there's a single force

applied to this particular ball ignore

air friction and all that kind of stuff

there's going to be a force of gravity

which is negative

m times g the negative is because my

position

my y here is defined to be positive

going up

and the force of gravity points down and

then it's the mass of the ball times

g the constant acceleration of gravity

in the surface of the earth

9.8 meters per second squared okay so

where's the differential equation

well it comes to us from newton's second

law

f equals mass times acceleration the sum

of the forces equaling the mass

times the acceleration so in this case

what is mass times acceleration

acceleration

is the second derivative of the position

function

so i'll write it this way the negative

mg that we saw before that's the only

force on it

is equal to m times the second

derivative of

y the acceleration is the second

derivative of the position function

y of t this again is a

differential equation now i've got a

second derivative in my equation

the m's on both sides so they cancel so

it's just y double prime is

a constant negative g now

this differential equation is one that i

can solve i can find the solution just

by integrating it's a second derivative

equal to a constant

so let's integrate both sides on the

left the negative g

turned into a negative g t and then i

add a plus

c because i'm doing an indefinite

integral and indefinite integrals give

you plus

c's and then on the right hand side the

y double prime turns into a y prime when

you integrate once

i integrate a second time the negative

gt

turns into a negative g t squared over 2

when you integrate once

the c integrates to a c t and then

because i'm doing a second integration

i get a second constant of integration

plus d

and then on the right hand side the y

prime turns into a y

so what i have here is just a solution

to this differential equation it's a

quadratic

and so as time goes on the ball is going

to go up and go down and this quadratic

behavior indeed

this is one of the kinematic equations

that you likely have seen before

what we've just done is to interpret it

in terms of

specifically a differential equation now

like the exponential growth example this

has two constants

a c and a d in it so how do i find the c

in the d and if you imagine when i

started with this ball

well the path of the ball would depend

on where my initial location that i

threw it from was

that would change the path of the ball

and it would also change the path of the

ball whether i gave

just a little bit of velocity to it

barely threw it up or whatever gave a

much higher amount

that would affect the path of the ball

so those initial conditions the initial

velocity and the initial position are

going to be relevant i'm going to state

them

i'll say my initial position at y equal

to zero is y naught

that's just a constant and y prime is

going to be

at t equal to zero v naught so y naught

and v naught are two constants

and if i plug t equal to zero into this

equation you just get

d is equal to y naught likewise if i

plug zero into the derivative of that

equation

i'm just going to get that the c is

equal to v naught

this leaves me my final solution to my

differential equation with the specific

choices

of the initial velocity and the initial

position specified

by the way it's going to be very

relevant to us in the future that in our

first example of exponential growth

we had one derivative which gave one

constant we had

one initial condition to specify it but

in this second example

we had a second derivative y double

prime which gave us two constants of

integration

and we needed two initial conditions to

specify them there's going to be a lot

of theory wrapped around

when exactly all this is going to work

out so let's leave with a few different

questions

first of all when do we actually have

solutions to differential equations if

you write some differential equation

some equation with a bunch of

derivatives in it when do they actually

have solutions

if they have a solution is it one or is

there

many different solutions i argued that

there were some today but maybe those

equations have

more and that's also something we'll

have to really investigate the theory

about

and then also how do you find solutions

in the first example i just asserted it

to you and the second we could compute

it but very often

we're going to have to learn a whole

bunch of procedures which says given a

differential equation

this is how we're going to try and solve

it so all that and more

coming up in the future videos so if you

enjoyed this video please do give it a

like

as i say this is part of a larger

playlist that consists of both the

videos

and the open source textbook the links

to both of those are down in the

description i very much appreciate you

sharing these resources with any other

classmates of yours who might be taking

differential equations and we're just

going to do some more math

in the next video