The first partitioning of the set of DEs is into the following two types: partial differential equations (PDEs) and ordinary differential equations (ODEs). A partial differential equation is one that involves any partial derivatives.

For example, the heat equation: \begin{equation} u_t = \alpha \nabla^2 u \qquad \text{OR} \qquad \frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \label{eqn-heat-pde}\tag{1.1.1} \end{equation} is an example of a PDE. This PDE models the flow or diffusion of heat in a three dimensional object. A long metal rod can be considered to be essentially a one–dimensional object. Heat flow in a rod would be governed by the one–dimensional heat equation: \begin{equation} \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}. \label{eqn-1d-heat-pde}\tag{1.1.2} \end{equation} Another important PDE is the wave equation: \begin{equation} u_{tt} = c^2 \nabla^2 u \qquad \text{OR} \qquad \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right). \label{eqn-wave-pde}\tag{1.1.3} \end{equation} A long thin wire is essentially a one–dimensional object. When plucked its vibrations are solutions to the one–dimensional wave equation first discovered by d'Alembert. \begin{equation} \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \label{eqn-1d-wave-pde}\tag{1.1.4} \end{equation} A solution to the one–dimensional wave equation must be a function of both position and time, \(u(x,t)\text{.}\)

An ordinary differential equation has only total derivatives. A total derivative of a function of one variable is just its normal derivative. For example if the position of an object is represented by \(x\text{,}\) then the total derivative of position with respect to time is written as: \(\frac{dx}{dt}\text{,}\) or \(x'(t)\text{.}\) Sir Isaac Newton was one of the first people to investigate ordinary differential equations. His famous second law of motion, \(F = ma\) is actually an ordinary differential equation, because if \(x(t)\) represents position as a function of time then acceleration is given by: \(a = \frac{d^2 x}{dt^2}\text{.}\) The famous equation, \(F=ma\) is thus written: \begin{equation} F = m\frac{d^2 x}{dt^2} \qquad \text{ OR } \qquad F = m x''. \label{eqn-newtons-2nd-law}\tag{1.1.5} \end{equation}

Polynomial equations are classified by their degree where the degree is the largest exponent which occurs in the polynomial. Typically, the higher the degree of the polynomial equation, the more difficult it is to solve. Like polynomials, differential equations can also be classified by a single positive integer.

Just like with polynomial equations of high degree, higher order DEs are in general more difficult to solve than lower order DEs. This is just a rule of thumb. In general, many factors contribute to whether a DE is solvable or not.

For example, the differential equation: \begin{equation} (y'')^3 + y' = y, \label{eqn-example-second-order}\tag{1.1.6} \end{equation} is an order two equation because the highest derivative in the equation is a second derivative. The fact that this term is cubed does not affect the order.

The notion of order applies to both ordinary and partial differential equations. Each example partial differential equation listed above is second order because the highest order derivative that appears in each equation is a second order partial derivative.

Every differential equation is either linear or nonlinear. This is perhaps the most useful categorization of DEs because although we know techniques for solving some (mostly first order) nonlinear equations, we have no general framework for understanding the solutions to such equations let alone systematic solution methods. Linear equations, are different. Linear algebra provides a framework for understanding solutions to linear ODEs. However, since we have yet to cover the basics of linear algebra, we will defer a proper definition of “linear” until chapter <<Unresolved xref, reference "chap-vector-spaces"; check spelling or use "provisional" attribute>>. For now we will content ourselves with a definition based upon the “form” of the ordinary differential equation. We will defer a discussion of linear PDEs to a later course.

Recall that an \(n^{th}\) degree polynomial equation will have \(n\) roots or solutions if we include repeated and complex roots. Solutions to linear ODEs are both different and analogous to the polynomial case. In general, a linear ODE will always have an infinite number of solutions, but all of these solutions can be constructed from \(n\) generator functions. An \(n^{th}\) order linear ODE will have \(n\) special generator functions which are all solutions to the ODE and which can be combined together to genarate all solutions.

Solving an equation can be difficult, but checking whether your solution is correct is relatively easy. For example, it is easy to check whether \(x=1\) is a solution to the polynomial equation: \begin{equation} x^3 - 4x^2 + 2x + 1 = 0. \label{eqn-simple-polynomial}\tag{1.1.8} \end{equation} To check you just substitute the value 1 for all occurrences of \(x\) in the equation. Doing so yields: \begin{align*} 1^3 - 4(1^2) + 2(1) + 1 &= 0,\\ 0 &= 0, \end{align*} a true statement, thus indicating that \(x=1\) is a solution to, or satisfies equation (1.1.8).

The situation with differential equations is similar except the solution to a differential equation is not a value, it is a function. Checking whether a function