First-Order Differential Equations
A first-order differential equation is a type of differential equation involving the variable \( x \), an unknown function \( y = f(x) \), and its first derivative \( y' = f'(x) \): $$ F(x, y, y') = 0 $$ In standard form, it can be rewritten as: $$ y' = G(x, y) $$
The goal is to determine the function \( y = f(x) \).
Solving such equations means finding a function \( f(x) \) whose derivative satisfies the equation.
A Practical Example
Consider the following first-order differential equation:
$$ F(x, y, y') = 0 $$
We're given the expression for the derivative:
$$ y' - 2x = 1 $$
We rewrite it in standard form as:
$$ y' = 1 + 2x $$
Our task is to find a function \( y = f(x) \) whose derivative is \( 1 + 2x \).
To do that, we compute the indefinite integral of \( 1 + 2x \):
$$ f(x) = \int (1 + 2x)\, dx $$
Which can be split as:
$$ f(x) = \int 1\, dx + \int 2x\, dx $$
$$ f(x) = x + x^2 + c $$
This gives us the general solution: a family of functions of the form \( f(x) = x^2 + x + c \).
For simplicity, we can omit the constant \( c \):
$$ f(x) = x^2 + x $$
Note: When a specific value is assigned to the constant \( c \), we obtain a particular solution to the differential equation. For instance: $$ f(x) = x^2 + x + 3 $$
Types of First-Order Differential Equations
First-order differential equations can be grouped into three main categories:
- Basic differential equations
These are the simplest type, typically written as: $$ y' = f(x) $$ - Separable differential equations
These involve functions that can be factored into separate terms in \( x \) and \( y \), taking the form: $$ y' = f(x)g(y) $$ - Linear differential equations
These consist of a linear combination of the unknown function and its derivative, such as: $$ y' + a(x)y = b(x) $$
And so on.
