1. Idea of a Differential Equation
A differential equation is an equation that contains one or more derivatives of an unknown function. These notes treat it as a way to describe how a quantity changes rather than the quantity itself. Many natural processes—growth, decay, motion, heating—are governed by rules of change, so differential equations express those rules in symbolic form.
The unknown in a differential equation is not just a number but an entire function. Solving the equation means finding a function whose derivatives fit the relation given.
1.1. Why differential equations feel natural
Whenever a relationship connects a quantity with its rate of change, a differential equation appears automatically. For example, if the rate at which something grows is proportional to its current amount, the equation involves the derivative of that amount. The symbol \(\frac{dy}{dx}\) simply represents the rate of change of \(y\) with respect to \(x\).
2. Formal Definition
A differential equation is an equation that involves an unknown function and one or more of its derivatives. The general form may be written as:
\(F\big(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots \big) = 0.\)
Here, \(F\) is any relation connecting the independent variable, the dependent variable, and its derivatives. The goal is to find a function \(y = y(x)\) that satisfies this relation.
2.1. Basic terminology used in differential equations
- Dependent variable: the quantity whose behaviour is described; often written as \(y\).
- Independent variable: the variable upon which the dependent variable relies; often \(x\).
- Derivative: a measure of how the function changes. \(\frac{dy}{dx}\) is the first derivative, \(\frac{d^2y}{dx^2}\) the second, and so on.
3. Structure of a Differential Equation
A differential equation may involve derivatives of various orders. The highest order of derivative that appears determines the nature of the equation. A few possible structures are:
- Equations involving only the first derivative.
- Equations involving up to the second derivative.
- Equations involving multiple derivatives and possibly both \(x\) and \(y\).
The equation can be linear, nonlinear, simple, or complex depending on how the derivatives appear.
3.1. Examples of structural forms
Below are a few example forms that reflect the variety of differential equations encountered:
- \(\frac{dy}{dx} = 3x\)
- \(\frac{d^2y}{dx^2} + y = 0\)
- \(x^2 \frac{d^2y}{dx^2} - 4x \frac{dy}{dx} + 6y = 0\)
- \(\left(\frac{dy}{dx}\right)^2 + y = x\)
Each of these contains at least one derivative, so each is a differential equation, even though their forms differ widely.
4. Simple Examples to Illustrate the Definition
These examples help make the idea of a differential equation concrete:
4.1. Example 1: A first-order equation
Consider the equation
\(\frac{dy}{dx} = y.\)
This states that the rate of change of \(y\) is proportional to \(y\) itself. The function that satisfies this relation must grow (or decay) in such a way that its derivative equals its own value.
4.2. Example 2: A second-order equation
Take the equation
\(\frac{d^2y}{dx^2} = -y.\)
This expresses a relationship between a function and its second derivative. Such equations often arise in oscillatory motion, where the acceleration is proportional to displacement but in the opposite direction.
5. Notes and Observations
Some useful points to keep in mind about differential equations:
- A differential equation always relates a function to its derivatives.
- The function to be found usually has infinitely many possible forms until extra information is given.
- Differential equations arise naturally whenever change, motion, growth, or decay is involved.
- The solution of a differential equation is not a number but a function.