What is a Differential Equation?
An equation involving an independent variable, a dependent variable, and the derivatives of the dependent variable with respect to the independent variable is called a differential equation.
Simply put — if an equation contains any derivative ($\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, etc.), it is a differential equation.
A differential equation is an equation of the form:
where $x$ is the independent variable and $y$ is the dependent variable.
Examples of Differential Equations
| Differential Equation | Description |
|---|---|
| $\dfrac{dy}{dx} = \cos x$ | First-order, rate of change equals $\cos x$ |
| $\dfrac{d^2y}{dx^2} + y = 0$ | Second-order, models simple harmonic motion |
| $x\,\dfrac{dy}{dx} + y = x^2$ | First-order linear differential equation |
| $\left(\dfrac{dy}{dx}\right)^2 + y = 0$ | Degree 2, first-order DE |
| $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} + 2y = e^x$ | Second-order linear with constant coefficients |
Real-World Applications
Differential equations are the language of nature. They describe how quantities change and are used extensively in:
Physics & Engineering
Newton's second law $F = ma$ translates to $m\dfrac{d^2x}{dt^2} = F(t)$. The motion of springs, pendulums, and electrical circuits are all governed by DEs.
Biology & Medicine
Population growth, spread of diseases, and drug concentration in the bloodstream are modeled using DEs. For example, the logistic growth model:
Economics & Finance
Continuous compound interest and the Black-Scholes model in finance are differential equations in disguise.
Types of Differential Equations
| Type | Meaning | Example |
|---|---|---|
| ODE | Ordinary DE — one independent variable | $\frac{dy}{dx} = x$ |
| PDE | Partial DE — more than one independent variable | $\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$ |
At Class 12 level, we study only Ordinary Differential Equations (ODEs).
🔑 Key Takeaways
- A differential equation involves a function and its derivatives.
- The independent variable is usually $x$ (or $t$); the dependent variable is $y$.
- DEs arise naturally when modeling real-world phenomena involving rates of change.
- Class 12 focuses exclusively on Ordinary Differential Equations (ODEs).
- Solving a DE means finding the function $y$ that satisfies it.