Courses

Differential Equations

Introduction
First Order Differential Equations
Wrońskian
Second Order Differential Equations
1. $a y^{″} + b y^{'} + c y = 0$
2. Spring-Mass-Damper
3. Method of Undetermined Coefficients
4. Reduction of Order
5. Variation of Parameters
6. Cauchy-Euler Equations
Laplace Transform
1. Lookup Table and WolframAlpha
Systems of ODEs
1. Simplest Case $[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$
  1. Poincaré Diagram
Preview of Dynamical Systems
1. Self-Study
2. Rössler Attractor

Additional Resources

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Notation and Definitions | Topic Playlist

What's a differential equation? | Explanation Video

A differential equation is an equation that has an unknown function and its derivatives present. The simplest example is $y^{'} (x) = y (x)$ with $y (x)$ an unknown function.

A non-example is $y^{'} (x) = x$ which is not usually considered a differential equation since $y (x)$ is not present.

Ordinary Differential Equation (ODE) vs. Partial Differential Equation (PDE) | Explanation Video

An ODE is a differential equation where the unknown function is a function of a single variable, like $y (x)$ .

A PDE is a differential equation where the unknown function is a function of a multiple variables, like $f (x, y)$ or $f (x, y, z)$ . An example is the temperature at every point $(x, y)$ on a flat metal plate or the density of air at every point $(x, y, z)$ in a music hall.

Different ways to write derivatives | Explanation Video

There's three common ways to write derivatives of a single variable function $y (x)$ . Which one to use is personal preference, but people usually pick the one that requires the least writing and is easiest to read.

\frac{d y}{d x} = y^{'} = y^{(1)} \frac{d^{2} y}{d x^{2}} = y^{″} = y^{(2)} \frac{d^{5} y}{d x^{5}} = y^{'''''} = y^{(5)}

A fourth way to write derivatives of a single variable function $u (t)$ is with the "dot notation" common in physics and engineering.

\frac{d u}{d t} = u^{'} = u^{(1)} = \dot{u} \frac{d^{2} u}{d t^{2}} = u^{″} = u^{(2)} = \ddot{u}

This notation is used for functions of time $t$ and you typically only see a single "u dot" or a "u double dot" since first and second time derivatives are common in physics and engineering. A differential equation with a "u triple dot" is uncommon.

There's two common ways to write partial derivatives for a function $f (x, y)$ . (Note: The order on the mixed partial changes between the two notations.)

\frac{\partial f}{\partial x} = f_{x} \frac{\partial^{2} f}{\partial x^{2}} = f_{x x} \frac{\partial f}{\partial y} = f_{y} \frac{\partial^{2} f}{\partial y^{2}} = f_{y y}

\frac{\partial}{\partial y} \frac{\partial}{\partial x} f = \frac{\partial^{2} f}{\partial y \partial x} = f_{x y} \frac{\partial}{\partial x} \frac{\partial}{\partial y} f = \frac{\partial^{2} f}{\partial x \partial y} = f_{y x}

Order of a differential equation | Explanation Video

The order of a differential equation is the highest derivative of the unknown function present.

\begin{aligned} y^{″} + y = 1 & 2nd order \\ (y^{'})^{5} + y = 1 & 1st order \\ y^{‴} y^{″} y^{'} y = x & 3rd order \\ \sin (y^{(4)}) + y = e^{x} & 4th order \end{aligned}

Linear vs. Nonlinear ODEs | Explanation Video

A linear differential equation has the following form.

p_{n} (x) y^{(n)} + p_{n - 1} (x) y^{(n - 1)} + \dots + p_{1} (x) y^{'} + p_{-} (x) y = g (x)

The functions of $x$ do not matter for the differential equation to be linear or nonlinear. The only thing that matters is the $y (x)$ function and all its derivatives multiply a function of $x$ . A differential equation is nonlinear means the differential equation does not fit the form of a linear differential equation.

\begin{aligned} y^{″} + y = 1 & linear \\ \tan (\frac{\ln (x)}{x^{2} + 1}) y^{″} + x^{x} y = e^{\sin (x)} & linear \\ (y^{'})^{2} + y = 7 & nonlinear \\ y^{″} y^{'} + y^{'} + x y = 31 & nonlinear \\ y^{(5)} + \frac{1}{y} = e^{x} & nonlinear \\ y^{″} + \sin (y^{'}) + y = \frac{1}{x} & nonlinear \\ y^{″} + e^{y} = 4 & nonlinear \\ \ln (y^{'}) + y = 0 & nonlinear \end{aligned}

Implicit vs. Explicit Solution | Explanation Video

An explicit solution to an ODE is $y (x) = f (x)$ . Ideally, we want to solve for $y$ in terms of $x$ , but that is not always possible.

For example, there is no way to solve for $y$ in the following equation.

y^{2} + e^{y} = \sin (x) + C \ln (x)

This is an implicit general solution for $x$ and $y$ .