Hello everybody! This is the first post for the differential equations tutorial. Unlike the one on MHF, I've decided to take this project a little more seriously and have taken on the task of making a book based on these posts (well, it's really the other way around - the posts will be the contents of the book with the exception of exercises that will be included at the end of each chapter/section).

For each chapter, a thread will be made and for each section, a post will be made. So for example, subsequent posts made in this thread will be material from the first chapter. Attached to each post will be the entire [updated] book as a .pdf (all nicely TeXed up and everything) so you can download it and have it for easy access.

I would also like to extend my thanks to Adrian for revising what I'm about to post and [hopefully] what I'll be posting in the future. In addition, I would also like to thank Fantini for volunteering to help revise future posts I make in this tutorial.

With all that said, let's get this thing started.

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Section 1: What Is a Differential Equation?

What is a differential equation? The adjective "differential'' seems to suggest that we will be dealing with derivatives...but derivatives of what? It turns out that a differential equation relates an unknown function with its derivatives. For a more formal definition, let us consider the following:

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Definition 1.1: Let $f$ be a function. Let $x:\mathbb{R}\rightarrow\mathbb{R}$ be an unknown differentiable function in variable $t\in\mathbb{R}$, and let $x^{\prime}(t),x^{\prime\prime}(t),\ldots,x^{(n)}(t)$ denote the derivatives of $x(t)$. Then the equation

\[f(t,x,x^{\prime},x^{\prime\prime},\ldots,x^{(n)}) = 0\qquad\qquad(1.1)\]

is called aordinary differential equation(which we will abbreviate as ODE). We define theorderof a differential equation to be the highest order derivative it contains.

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Definition 1.2: Let $u:\mathbb{R}^{n}\to\mathbb{R}$ be an unknown differentiable function of the multiple independent variables $x_{1},\dots,x_{n}$. Then the equation

\[F\left(u,x_{1},\dots,x_{n},\frac{\partial u}{\partial x_{1}},\dots,\frac{\partial u}{\partial x_{n}},\frac{\partial^{2}u}{\partial x_{1}\partial x_{1}},\dots,\frac{\partial^{2}u}{\partial x_{1}\partial x_{n}},\dots\right)=0\qquad\qquad(1.2)\]

is called apartial differential equation(which we will abbreviate as PDE). Again, the order of the partial differential equation is the order of the highest derivative appearing in the partial differential equation.

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By the above definition, (1.1) would be an $n$-th order ODE since $x^{(n)}(t)$ is the highest order derivative of $x(t)$. Likewise, (1.2) would be an $n$-th order PDE. ODEs and PDEs are the two most common types of differential equations. Both types of differential equations are useful in their own ways, butwe will focus only on ODEs for the time being(the first part of this book/tutorial will be on ODEs only; part 2 will cover topics in PDEs).

Within each of these main classes, we can further categorize a differential equation by determining whether or not an ODE or PDE islinearornon-linear. An ODE is said to be linear if the function $f$ is a linear function in $x(t)$ and its derivatives. A function $f:\mathbb{R}\to\mathbb{R}$ is linear if for all $c,x,y\in\mathbb{R}$, it is true that $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$. A linear ODE can be written as

\[a_{n}(t)\,x^{(n)}+a_{n-1}(t)\,x^{(n-1)}+\dots+a_{1}(t)\,x'+a_{0}(t)\,x=g(t).\qquad \qquad(1.3)\]

A PDE is linear if the function $F$ is a linear function of $u$ and all its derivatives.

Otherwise, the ODE or PDE is said to be non-linear. To get a better understanding of what this means, let's look at a few examples.

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Example 1.1(Harmonic Oscillator): The differential equation

\[\frac{d^2x}{dt^2}+\omega^2x = 0\qquad\qquad(1.4)\]

is a second order linear ODE. This models the motion of a spring mass system without any damping factors.

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Example 1.2(Two-dimensional Wave Equation): The differential equation

\[\frac{\partial^2 u}{\partial t^2} = c^2\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)\qquad\qquad(1.5)\]

is a second order linear PDE. This models the vibration of a membrane.

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Example 1.3(Navier-Stokes Equations): The differential equation

\[\rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla p + \mu\nabla^2 + \mathbf{f}\qquad\qquad(1.6)\]

is a second order non-linear PDE. It's second order due to the fact that

\[\nabla^2 = \nabla\cdot\nabla = \sum_{i=1}^n\frac{\partial^{2}}{\partial x_i^2}\]

in $\mathbb{R}^n$. This is the Navier-Stokes equation for an incompressible fluid, and is the subject of one of the seven Millennium Prize Problems proposed by the Clay Mathematics Institute.

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Example 1.4(Motion of a Pendulum): The differential equation

\[\frac{d^2u}{dt^2}+\frac{g}{L}\sin(u) = 0\qquad\qquad(1.7)\]

is a second order non-linear ODE. This models the motion of a pendulum with length $L$.

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Now that we have a better understanding for what a differential equation is, we should now discuss a special type of ODE or PDE -- theinitial value problem.

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Definition 1.3(Initial Value Problem): Aninitial value problem(abbreviated as IVP) is a differential equation coupled with aninitial condition$f(x_0)=y_0$, where $f$ is the solution of our ODE (or PDE) with $x_0\in\mathbb{R}^n$ and $y_0\in\mathbb{R}$.

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The thing that makes an IVP different from the general ODE or PDE is that a solution of the IVP is said to be aparticular solution(i.e., the solution contains no arbitrary constants that may arise from techniques needed to solve the equation). A solution of an ODE or PDE (given that one exists to begin with) with no initial conditions is called ageneral solution. For example, if we have the differential equation

\[\frac{dy}{dx}+y=x,\qquad\qquad(1.8)\]

it's general solution is $y(x)=x-1+Ce^{-x}$. However, if we couple it with the initial condition $y(0)=0$, we then end up with the particular solution $y(x) = x-1+e^{-x}$. For various initial conditions, however, the solution to an ODE or PDE may or may not exist! This leads into the next important topic -- existence and uniqueness of solutions to differential equations. For now, we will restrict ourselves to the case of first order ODEs. We will state the theorem on existence and uniqueness of solutions, but will continue the discussion of existence and uniqueness in the next section, once we have a means for solving simple ODEs.

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Theorem 1.1(Existence and Uniqueness of Solutions): Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a differentiable function with partial derivative $f_y$. Suppose that both $f(x,y)$ and $f_y(x,y)$ are continuous on some rectangle $R\subset\mathbb{R}^2$ such that for some $(a,b)\in\mathbb{R}^2$, we have $(a,b)\in R$. Then, for some open interval $I$ containing the point $a$, the IVP

\[\frac{dy}{dx} = f(x,y),\quad y(a) = b\qquad\qquad(1.9)\]

has exactly one solution that is defined on the interval $I$.

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We should now have a much better understanding for what differential equations are and what their various characteristics are. We will now start the long journey of understanding how to solve different kinds of differential equations. I hope you're ready to march forward with me!

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Stay tuned for the second post, which deals with equations of the form $\dfrac{dy}{dx} = f(x)$ (you can currently see part of it done in the file, but I plan to make changes to it...so yea).

Comments and questions should be posted here: