Introduction to Differential Equations
A differential equation is an equation that relates a function to its derivatives. These powerful mathematical tools describe how things change — from population growth to the motion of planets.
📐 Formal Definition
A differential equation (DE) is an equation that contains one or more derivatives of an unknown function.
The most general form is:
$$ F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots, \frac{d^ny}{dx^n}\right) = 0 $$
where $y = f(x)$ is the unknown function we want to find.
Key Insight: In algebra, you solve for a number. In differential equations, you solve for a function.
🔑 Key Terminology
| Term | Meaning |
|---|---|
| Order | The highest derivative that appears in the equation |
| Degree | The power of the highest-order derivative (when polynomial in derivatives) |
| ODE | Ordinary Differential Equation — involves one independent variable |
| Solution | A function $y = f(x)$ that satisfies the equation |
| General Solution | Contains arbitrary constants (a family of curves) |
| Particular Solution | A specific solution with constants determined by initial conditions |
📊 Examples of Differential Equations and Their Orders
First Order (Order 1)
$$ \frac{dy}{dx} = 3x + 2 $$
$$ \frac{dy}{dx} = ky $$
Highest derivative: $\frac{dy}{dx}$Second Order (Order 2)
$$ \frac{d^2y}{dx^2} + 4y = 0 $$
$$ \frac{d^2y}{dx^2} = -9.8 $$
Highest derivative: $\frac{d^2y}{dx^2}$Third Order (Order 3)
$$ \frac{d^3y}{dx^3} + x\frac{dy}{dx} = e^x $$
Highest derivative: $\frac{d^3y}{dx^3}$