The Key Definitions of Differential Equations: ODE, order, solution, initial condition, IVP

Summary

This educational video delves into the fundamentals of differential equations, introducing the concept with a specific example and explaining the terminology of ordinary versus partial differential equations. It emphasizes the importance of the order of a differential equation and illustrates how to identify and verify solutions, including the use of technology like the Maple Calculator app. The script also touches on initial value problems, showcasing how to find specific solutions given certain conditions, and encourages viewers to explore the fascinating world of differential equations.

Takeaways

• 📚 The video is part of a series on differential equations, introducing major concepts and providing access to a full course and open source textbook.
• 🔍 The script provides a formal definition of a differential equation, emphasizing its components such as variables, derivatives, and the relationship between them.
• 📉 The concept of 'order' in differential equations is introduced, highlighting that it refers to the highest derivative present in the equation.
• 📈 The difference between ordinary and partial differential equations is explained, with the former involving a single independent variable and the latter involving multiple variables and partial derivatives.
• 🔑 The script discusses the idea of solving differential equations, illustrating how certain functions can satisfy a given differential equation and be considered solutions.
• 🔍 The process of finding solutions to differential equations is mentioned, with the promise of exploring methods in later videos of the course.
• 🌐 The use of technology, specifically the Maple calculator app, is showcased as a tool for solving differential equations by interpreting handwritten or typed equations.
• 📝 The script introduces the concept of a general solution, which contains all possible solutions to a differential equation, typically represented with arbitrary constants.
• 🔑 The importance of initial conditions in defining a specific solution to a differential equation is highlighted, transforming a general solution into a particular one.
• 📈 The notion of an initial value problem is explained, which involves a differential equation and specific initial conditions to find a unique solution.
• 📚 The video concludes by emphasizing the foundational terminology and concepts introduced, setting the stage for further exploration in the course.

Q & A

What is the main topic of the video?

The main topic of the video is differential equations, focusing on introducing the major ideas and concepts related to this mathematical field.

What is the difference between an ordinary differential equation and a partial differential equation?

An ordinary differential equation involves derivatives with respect to a single independent variable, while a partial differential equation involves partial derivatives with respect to multiple variables.

What is the order of a differential equation?

The order of a differential equation is the highest derivative that appears in the equation. For example, if the highest derivative is the third derivative, the order is 3.

What is the purpose of solving a differential equation?

The purpose of solving a differential equation is to find the function or functions that satisfy the given equation, which can describe various phenomena in science and engineering.

How does the video illustrate the concept of a solution to a differential equation?

The video illustrates the concept by providing examples of functions, such as e^t and c1e^t + c2e^(3t), that when substituted into the differential equation, satisfy the equation, thus qualifying as solutions.

What is the general solution to a differential equation?

The general solution to a differential equation is an expression that contains every possible solution to the equation, often involving arbitrary constants that can take on different values.

What is an initial value problem in the context of differential equations?

An initial value problem is a differential equation supplemented with initial conditions, which are specific values of the function and its derivatives at a given point, usually at t=0.

How does the video mention the use of technology to solve differential equations?

The video mentions the use of the Maple Calculator app, which can interpret handwritten or typed differential equations and provide the general solution.

What is the significance of the arbitrary constants in the general solution of a differential equation?

The arbitrary constants in the general solution allow for the representation of an infinite family of solutions, as they can take on any value, thus capturing all possible solutions to the equation.

How does the video define the solution to an ordinary differential equation?

The video defines a solution to an ordinary differential equation as a specific function that, when substituted into the equation, satisfies the equation for all values of the independent variable.

What is the role of initial conditions in solving an initial value problem?

Initial conditions in an initial value problem provide specific values for the function and its derivatives at a certain point, which helps in determining the particular solution that satisfies both the differential equation and the initial conditions.

Outlines

00:00

📚 Introduction to Differential Equations

The video script begins with an introduction to differential equations, focusing on the major concepts of the topic. It is the second video in a playlist and course dedicated to this subject. The speaker provides a link to the course and an open source textbook in the description. The script clarifies the definition of a differential equation, using an example that includes variables, derivatives, and a relationship between them. It distinguishes between ordinary and partial differential equations, highlighting the importance of the order of the equation, which is the highest derivative present. The video promises to start with first-order differential equations and then progress to higher orders, with a brief mention of systems of differential equations used in models like the S-I-R model for pandemics.

05:00

🔍 Exploring Solutions to Differential Equations

This paragraph delves into the process of solving differential equations, starting with an example equation and demonstrating how to identify solutions. The script introduces the concept of the general solution, which encompasses all possible solutions to a given differential equation, expressed as combinations of specific functions and constants. It illustrates this with examples of solutions involving exponential functions and emphasizes the linearity of derivatives, which allows for the summation of solutions. The speaker also mentions the use of technology, specifically the Maple calculator app, to assist in finding solutions by simply photographing or typing in the equation. The paragraph concludes with a discussion on the theory of solutions and their connection to the order of the differential equation.

10:02

📉 Initial Value Problems and Specific Solutions

The final paragraph of the script discusses initial value problems, which are differential equations accompanied by initial conditions specifying the values of the function and its derivatives at a particular time, usually t=0. The speaker explains how to solve for a specific solution that satisfies both the differential equation and the initial conditions. An example is given where an initial condition is applied to a general solution to find a unique solution. The importance of providing n initial conditions for an nth order differential equation is highlighted. The video concludes with an invitation for viewers to engage with the content by liking the video, asking questions, and looking forward to further exploration of differential equations in subsequent videos.

Keywords

💡Differential Equations

Differential equations are mathematical equations that involve derivatives, which represent rates of change. In the context of the video, differential equations are the central theme, with the speaker aiming to formalize and explain what they are. The video provides an example of a differential equation involving the third derivative of 'y' with respect to 't', illustrating the concept.

💡Ordinary Differential Equations (ODEs)

ODEs are a type of differential equation that involves derivatives with respect to a single independent variable. The video script introduces ODEs and distinguishes them from partial differential equations by noting that they involve only one independent variable, 't' in the given examples.

💡Derivatives

Derivatives in calculus represent the rate at which a quantity changes with respect to another quantity. In the video, derivatives are the core of differential equations, with the script mentioning 'y prime', 'y double prime', and 'y triple prime' to denote the first, second, and third derivatives of 'y' with respect to 't'.

💡Order of a Differential Equation

The order of a differential equation is defined by the highest derivative present in the equation. The script uses the term to explain that the differential equation with a 'y triple prime' has an order of 3, which is a key characteristic in classifying and solving differential equations.

💡Partial Differential Equations (PDEs)

PDEs are introduced in the script as a category of differential equations that involve multiple variables and partial derivatives. The script contrasts PDEs with ODEs by showing an example of Schrödinger's equation from quantum mechanics, which involves both 'x' and 't' and their respective partial derivatives.

💡Solutions to Differential Equations

In the context of the video, a solution to a differential equation is a function that, when substituted into the equation, satisfies the equation. The script provides examples of solutions, such as 'e to the t' and 'c1 times e to the t', and discusses how these can form a general solution to the equation.

💡General Solution

A general solution to a differential equation includes all possible solutions, typically involving arbitrary constants. The script explains that the general solution to the given differential equation can be written as a combination of 'c1 times e to the t' and 'c2 times e to the 3t', encompassing an infinite family of solutions.

💡Initial Value Problem (IVP)

An IVP is a differential equation supplemented with initial conditions, which are specific values of the function and its derivatives at a given point. The script introduces the concept by providing an initial condition for the function 'y' at 't equals 0', and shows how it can be used to find a specific solution from the general solution.

💡Maple Calculator

The Maple Calculator, mentioned as the first official sponsor of the math channel in the script, is an app that can solve differential equations. The video demonstrates its use by showing how to input a differential equation, either by handwriting or typing, and obtaining the general solution, as exemplified with the equation 'y double prime - 4y prime + 3y equal to 0'.

💡Linear Operators

Linear operators are a concept from linear algebra that are relevant to differential equations because the operations of taking derivatives and adding functions are linear. The script briefly touches on this by stating that the sum of solutions to a differential equation is also a solution, which is a property of linearity.

💡Systems of Differential Equations

A system of differential equations involves multiple differential equations that are interrelated. The script mentions this concept in the context of the S-I-R model for studying pandemics, indicating that it is a more complex scenario where multiple equations are involved, each potentially representing different aspects of the system.

Highlights

Introduction to the major ideas of differential equations in the second video of the playlist.

Formal definition and explanation of what a differential equation is, including an example with y triple prime and t y prime sine of y double prime.

Differentiation between ordinary differential equations (ODEs) and partial differential equations (PDEs), with an example of Schrödinger's equation.

Explanation of the order of a differential equation, defined by the highest derivative present.

Introduction to the concept of solving a differential equation and the process of verifying a solution.

Demonstration of how constants can affect the solutions of a differential equation, with the example of e^t and c1e^t.

Introduction of the general solution concept, encompassing all possible solutions to a differential equation.

Use of technology, specifically the Maple Calculator app, to solve and verify differential equations.

Definition of a solution to an ordinary differential equation and the process of assigning a specific function to solve it.

Example of a first-order differential equation and its general solution involving e to the power of t cubed.

Discussion on the importance of the constant 'c' in solutions and how it can be determined with additional information.

Introduction to initial value problems and how they differ from general solutions by requiring specific values for the function and its derivatives at a given time.

Explanation of how to solve an initial value problem, including the need for n initial conditions for an nth order differential equation.

Highlighting the process of finding a specific solution that satisfies both the differential equation and the initial conditions.

Foundational terminology and concepts established for further study in differential equations.

Invitation for viewers to engage with the content by liking the video and leaving comments with questions.

Transcripts

in this video

we're going to talk about differential

equations and i want to

introduce the major ideas of this topic

this is actually the

second video in my playlist and my full

course on differential equations

the link to that and the accompanying

open source textbook is down in the

description

now in the first video i just gave a

loose introduction to what differential

equations were about but in this video

i really want to formalize precisely

what it is that we're talking about

so let's begin with an example of a

differential equation

y triple prime plus t y prime sine of

y double prime equal to zero so this is

an equation it's got both sides

it's got some variable t and derivatives

of

y with respect to t it's a differential

equation

because it's an equation involving

derivatives to be a little bit more

precise

we're going to begin by talking about

ordinary differential equations

and in this description on the left hand

side i have the nth derivative of y

when you put the n in brackets and put

it as a superscript

this denotes the nth derivative not to

be confused with

y to the power of n this is the nth

derivative of y

with respect to t and then on the right

hand side i have some relationship some

function that i haven't specified

between a dependent variable that i'll

call t

sometimes t sometimes x and a dependent

variable

y and its derivatives y y prime y double

prime

all the way potentially up to y to the n

minus one derivatives and so your nth

derivative with respect to t

depends on the previous derivatives the

lower order derivatives and the

independent variable

t now differential equations have

different types of properties and one of

them is referred to as

the order of a differential equation and

the

order of a differential equation it's

just the highest derivative that appears

so in my example where there was a y

triple prime

that y triple prime was the highest

order of a derivative and so we say

the order of that differential equation

is 3

and in the general case that is an nth

order differential equation the

highest derivative that appears is the

nth derivative

and indeed order is going to be a

distinguishing feature for us in our

study of differential equations we're

going to begin by studying

first order differential equations where

there's only one derivative involved

this is going to take actually quite a

bit of development and then we'll move

to higher order differential equations

after that

now you'll notice that i call this an

ordinary differential equation

so what are the other options now we're

not going to talk about these until

quite a long time from now but the other

big category is so-called partial

differential equations

in this equation schrodinger's equation

from quantum mechanics one of the most

important differential equations that

there is

we have some function that depends on

both x and t

and then you have derivatives with

respect to t and derivatives respect to

x they're

partial derivatives now so when your

differential equation

is involving multiple variables and

they're partial derivatives

then we call it a partial differential

equation if there's just

one independent variable then it's an

ordinary differential equation and

indeed

we're going to be focusing on ordinary

differential equations at least for now

the other complexity that can happen and

again this is going to be something for

the future

is systems of differential equations

where you have not

one but multiple different differential

equations these

are the equations for the s-i-r model

that helps us study pandemics

i actually have a couple of videos on

this model this system of differential

equations so you're welcome to check

that out if you so wish

regardless our focus is going to be on

ordinary differential equations

like this one okay so now that you have

a differential equation what can you do

with it well the main thing we're trying

to do is to

solve a differential equation so what

exactly does that mean

take for instance the function e to the

t i'll call this

y one because this is going to be the

first of multiple solutions

to this differential equation my claim

is that e to the t

solve this equation how do i know well

i'm going to plug it in and i'm going to

see that if i plug that in

so two derivative t is just to the t

then i subtract off

one derivative of e to the t times four

it's just

four e to the t and then plus three

times plug it in e to the t

and indeed e to the t minus four e to

the t

plus three e to the t yeah that adds up

to zero so yes

this satisfies the differential equation

that is i

take this function i plug it into the

differential equation and if it

satisfies it then we're going to call it

a solution

but i want you to note that in addition

i could put some

other constant c out the front of it

some c

one because this is the first function i

talk about c one e the t

and the equation is exactly the same i

mean constants just come out of

derivatives so

it's just a c one in front of every term

and so that adds up to zero

in just the same way so we already have

an infinite family of solutions but it's

actually even more complicated

than that consider what i'll now call

y2 which is some constant times e to the

3t no longer e to the t e to the 3t well

it's

also a solution indeed plug it in the

same way

and you're going to get 9 times the

constant e to the 3t

that minus 12 the constant e to the 3t

and then plus 3 the constant times e to

the 3t which adds up to 0

it's again a solution so we got a y1 and

a y2

and then i'll actually leave it to you

to verify if you wish that any

combination like if you add those two

that is also going to be a solution this

is because derivatives are

linear operators is one way to put it so

the sum of them is going to result in

the sum of the solutions 0

0 is just going to be 0. and

now i've gotten something that i'm going

to isolate and refer to as

the general solution the general

solution

contains every possible solution to this

differential equation

that is any possible solution to this

differential equation can be written

as the combination of c1 times the e to

the t and

c2 times e to the 3t for different

values of those constants

but okay there's a lot of questions here

like how did i know what those solutions

were

and how do i know that all of the

solutions can be written in this way

perhaps there's

many other solutions that i haven't

written down now

to the question of how do you find these

solutions like how did i know e to the t

and e to the 3t

this is a subject of our course we're

going to study that later on in later

videos in this course

but for now we can actually use

technology to help us out

and the actual technology we're going to

use comes from and i'm

very proud to say it the first official

sponsor

of this math channel which is maple what

i can do with

maple calculator which is an app that

you can install on your phone and the

links are down in the description

is i can just hand write out the

differential equation that i'm trying to

study so i

can hand write out y double prime minus

4y prime plus 3y equal to 0.

then if i open the app i can hit the

camera button and i can just take a

picture of my equation

and the maple calculator will interpret

it and look what it does

it automatically spits out the general

solution to this differential equation

if you didn't want to bother with taking

a photo it's also totally fine for you

to type it in and to change the

constants and see what the solutions are

to any differential equation that you're

interested in

and it's nice to know that according to

the maple calculator this is

all of the solutions this is the general

solution

indeed there's going to be a lot of

theory that will develop later in the

course as to

how many solutions a different type of

equation can happen it can be very

intimately connected

to the order of the differential

equation okay so let's define this a bit

more precisely

now i want to talk about the solution to

an ordinary differential equation so

i'll put up my

generic ordinary differential equation

and then a solution

to an ordinary differential equation is

you telling me a specific function

i'm calling it here for example phi of t

and i'm saying that

i'm going to assign y to be that

specific function and the specific

function when you plug it in everywhere

it's the same equation

just to put phi everywhere instead of y

it solves that equation

so that's our precise definition of a

solution i'm going to give a second

example of a differential equation now

y prime equals y times t squared and

this is just another differential

equation first order and it has a

general solution

y of t is a constant times e to the t

cubed

divided out by three so in this case

one infinite family and again i'm not

going to go into how i solve this but

you can

verify that as a solution by taking the

derivative plugging it in

and seeing that it works now i really

want to focus on that c

value so i want to give an extra piece

of information

suppose i told you that at time t equal

to 0

y was 5 a so called initial condition

i'm telling you what happens

initially at time t equal to 0. well i

could plug this in

so that would be 5 on the left-hand side

equal to a constant e to the 0

if i plug in 2 equal to 0 e to the 0 is

1 and so i get that

c was equal to 5. and this lets me

rewrite what was originally a general

condition with an arbitrary constant i

now get a very specific answer y

of t is 5 e to the t cubed

divided by 3. this is referred to as an

initial

value problem more generally i would

start with my differential equation as i

have

but then in addition i would specify

initial conditions

what you actually have to do is if you

have an nth order differential equation

so

n derivatives you have to specify n

initial conditions

y 0 y prime is 0 all the way down to the

n minus 1 derivative

of y at 0 and you specify the values of

those

when i say y of 0 is y naught why not

it's just my

shorthand for some constant that will be

specified y naught

prime is some constant that will be

specified again whenever i write this

little subscript zero

i call it not and i just mean by that it

is a constant

so i have these initial conditions i

just tell you the values

of the function and its derivatives at

time t equal to zero and collectively

that is known as an initial value

problem a differential equation

together with initial conditions and to

solve

an initial value problem you're trying

to find a solution

that is not a general solution it

doesn't have constants in it doesn't

express

all the possible solutions to it now

we're solving

for the specific solution that not only

satisfies the differential equation but

also satisfies all of these initial

conditions

so in this video i hope i have

introduced the major definitions and

delineations between concepts that we're

going to see in differential equations

of course there's a lot more to talk

about we haven't even figured out our

first method

to solve a differential equation but at

least we've established our foundational

terminology

as we go forward in this course so if

you like this video please do give it a

like

for the youtube algorithm we need to

spread differential equations to

everybody it is such a cool subject

if you have any questions leave them

down in the comments below and we're

going to do some more math

in the next video