We use equations to describe relationships between quantities.
Often, some quantities are known, while others are unknown and need to be determined.
We need to “solve for” the unknown quantities in terms of the known ones.
Equations with numbers and algebraic operations
For example, in the equation \[ 2x + 3 = 9, \] we know the numbers \(2\), \(3\), and \(9\), the algebraic operations connecting them, and we want to solve for the unknown number \(x\).
The answer is easy: \(x = 3\).
Equations with functions and algebraic operations
For example, in the equation \[ 2f(x) + 4 = 10x^2, \] we know the numbers \(2\), \(4\), and the function \(10x^2\), the algebraic operations connecting them, and we want to solve for the unknown function \(f(x)\).
The answer is also easy: \(f(x) = 5x^2 - 2\).
Equations with functions and differential operations
For example, in the equation \[ \frac{dy}{dx} = 2y, \] the function \(y(x)\) is unknown. It is embedded in an equation that involves an algebraic operation (multiplication by \(2\)) and a differential operation (taking the derivative with respect to \(x\)).
The answer is a bit less easy: \(y(x) = Ce^{2x}\), where \(C\) is an arbitrary constant determined by initial (or boundary) conditions.
Such equations are called differential equations because they involve differential operations.
This one happens to be an ordinary differential equation (ODE) because the unknown function \(y\) depends on a single variable \(x\).
ODEs appear everywhere in the real world.
Example: Newton’s second law of motion
This equation says that \[ F = m \frac{d^2x}{dt^2}, \]
where \(x(t)\) is the position of an object as a function of time \(t\), \(m\) is its mass, and \(F\) is the force acting on it.
Definition
A partial differential equation (PDE) is a differential equation that involves an unknown function of multiple variables and its partial derivatives.
Example: the \(1\)-dimensional wave equation
This equation says that
\[ \frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2}, \]
where \(u(x,t)\) is a two-variable function and \(a\) is a known constant. It models wave-like phenomena, such as vibrations of a string, sound waves, electromagnetic waves, etc.
Example: the \(2\)-dimensional wave equation
This equation says that
\[ \frac{\partial^2 u}{\partial t^2} = a^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right), \]
where \(u(x,y,t)\) is a three-variable function and \(a\) is a known constant. It models wave-like phenomena in two spatial dimensions, such as waves on the surface of a pond.