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Differential Equations

The language of the universe: Predicting how systems change.

In algebra, we solve equations to find a number (e.g., x² + 2x = 8, so x = 2). In Differential Equations (DEs), the answer isn't a number—it is a function.

A differential equation relates a function to its own derivative. In simple terms, it relates the "current state" of a system to "how fast the system is changing".

1. What is a Differential Equation?

Imagine a population of rabbits. The more rabbits you have, the faster they breed. This means the rate of growth (derivative) depends on the current population size.

The Equation:
dP/dt = k · P

(Rate of change of Population = constant × Current Population)

To solve this, we need to find a function P(t) that makes this statement true. (Spoiler: It's exponential growth!).

[Image of exponential growth curve vs linear growth]

2. Types of Differential Equations

  • Ordinary Differential Equations (ODEs): Depend on only one variable (like time). Example: A swinging pendulum.
  • Partial Differential Equations (PDEs): Depend on multiple variables (like time and space). Example: Heat spreading across a metal plate.

3. Solving by Separation of Variables

One of the most common methods for solving basic ODEs is "Separation of Variables". The goal is to get all the y's on one side and all the x's on the other, and then integrate both sides.

Example: dy/dx = x/y

Step 1: Separate
Multiply by y and by dx:
y dy = x dx

Step 2: Integrate
∫ y dy = ∫ x dx
y²/2 = x²/2 + C

Step 3: Solve for y
Multiply by 2:
y² = x² + 2C (Let's call 2C a new constant, K)
y = ±√(x² + K)

4. Slope Fields

Sometimes, we can't solve the equation perfectly. However, we can visualize it using a Slope Field. Since the differential equation gives us the slope (dy/dx) at every point (x, y), we can draw tiny tangent lines on a grid.

By following these lines, you can trace the path of the solution curve without doing any algebra.

5. Real World Applications

Differential equations are used everywhere in science and engineering:

  • Newton's Law of Cooling: Describes how a hot cup of coffee cools down to room temperature.
  • Spring-Mass Systems: Describes how shock absorbers works in a car (Damped Harmonic Motion).
  • Epidemiology: The SIR model uses differential equations to predict how a virus spreads through a population.