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Chapter 5 Differential Equations

Many physical phenomena can be modeled using the language of calculus. For example, observational evidence suggests that the temperature of a cup of tea (or some other liquid) in a room of constant temperature will cool over time at a rate proportional to the difference between the room temperature and the temperature of the tea.

In symbols, if \(t\) is the time, \(M\) is the room temperature, and \(f(t)\) is the temperature of the tea at time \(t\) then \(f'(t) = k(M-f(t))\) where \(k>0\) is a constant which will depend on the kind of tea (or more generally the kind of liquid) but not on the room temperature or the temperature of the tea. This is Newton's law of cooling and the equation that we just wrote down is an example of a differential equation. Ideally we would like to solve this equation, namely, find the function \(f(t)\) that describes the temperature over time, though this often turns out to be impossible, in which case various approximation techniques must be used. The use and solution of differential equations is an important field of mathematics, because differential equations help us to predict future behaviour based on how current values are related and how they change with respect to each other (perhaps over time). Here we see how to solve some simple but useful types of differential equation.

Informally, a differential equation is an equation in which one or more of the derivatives of some function appears. Typically, a scientific theory will produce a differential equation (or a system of differential equations) that describes or governs some physical process, but the theory will not produce the desired function or functions directly.

Definition 5.1.

{Differential Equation} A differential equation is a mathematical equation for an unknown function of one (or several) variables that relates the function to its derivatives.

A solution to a differential equation is a function that satisfies the differential equation.

Note: The term Differential Equation is often abbreviated with DE, and so DEs stands for differential equations.

The following are examples of differential equations:

  1. \(y' = e^x\sec y\)

  2. \(y'-e^xy+3 = 0\)

  3. \(y'-e^xy = 0\)

  4. \(3y''-2y'=7\)

  5. \(4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0\)

Clearly, there are many different characteristics of a differential equation. The characteristics that are used throughout the notes are introduced below. However, there are additional ways to classify differential equations, which we leave to the interested reader, who pursues this field of study.