Differential Equations

Differential Equations

Part 1 of 7 — Introduction to Differential Equations

What is a Differential Equation?

A differential equation (DE) is an equation involving a function and its derivative(s).

Examples:

• $\frac{dy}{dx} = 3x^2$ — directly integrable
• $\frac{dy}{dx} = 2y$ — the rate depends on the current value
• $\frac{dy}{dx} = x + y$ — non-separable (not on AP AB)

Solving by Direct Integration

$\frac{dy}{dx} = f(x) \implies y = \int f(x)\,dx$

Worked Example

$\frac{dy}{dx} = 6x^2 - 4x + 1$, .

$y = 2x^3 - 2x^2 + x + C$

$y(0) = C = 3$. So $y = 2x^3 - 2x^2 + x + 3$.

Direct Integration 🎯

Key Takeaways — Part 1

1. A DE relates a function to its derivatives
2. Direct integration works when $\frac{dy}{dx} = f(x)$
3. Use initial conditions to find $C$

Differential Equations

Part 2 of 7 — Separation of Variables

The Method

For DEs of the form $\frac{dy}{dx} = f(x) \cdot g(y)$:

Differential Equations

Part 3 of 7 — Slope Fields

What is a Slope Field?

A slope field (direction field) is a visual representation of a DE. At each point $(x, y)$, a small line segment shows the slope $\frac{dy}{dx}$.

Differential Equations

Part 4 of 7 — Exponential Growth and Decay

The Model

$\frac{dy}{dt} = ky$

Solution: $y=y_0{e}^{}$

Differential Equations

Part 5 of 7 — More Separation of Variables Practice

Harder Examples

$\frac{dy}{dx} = \frac{y^2}{x}, \quad y(1) = 2$

Differential Equations

Part 6 of 7 — AP-Style Workshop

AP-Style DE Problems 🎯

Workshop Complete!

Differential Equations — Review

Part 7 of 7 — Comprehensive Assessment

Final Assessment 🎯

Differential Equations — Complete! ✅