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

$\frac{dy}{dx} = 6x^2 - 4x + 1$, $y(0) = 3$.

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

$y(0) = C = 3$. 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 $\frac{dy}{dx} = f(x) \cdot g(y)$:

1. Separate: $\frac{dy}{g(y)} = f(x)\,dx$
2. Integrate both sides
3. Solve for $y$ (if possible)

Worked Example

$\frac{dy}{dx} = xy$, $y(0) = 2$.

$\frac{dy}{y} = x\,dx$

$\ln|y| = \frac{x^2}{2} + C$

$y = Ae^{x^2/2}$ where $A = e^C$

$y(0) = A = 2$. So $y = 2e^{x^2/2}$.

Separation of Variables 🎯

Key Takeaways — Part 2

1. Separate variables: get all $y$ on one side, all $x$ on the other
2. Integrate both sides
3. Don't forget the constant $+C$ (or $A = e^C$ for exponentials)

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 $\frac{dy}{dx}$.

Reading Slope Fields

Matching Slope Fields to DEs

To match a slope field to a DE:

1. Check specific points: what's the slope at $(0,0)$, $(1,1)$, etc.?
2. Look for where slopes are zero (horizontal segments)
3. Look for patterns (same slopes on lines, etc.)

Slope Field Analysis 🎯

Key Takeaways — Part 3

1. Slope fields visualize the behavior of solutions
2. Solutions follow the slope field like flowing water
3. Check where slopes are 0, positive, or negative to match DEs

Differential Equations

Part 4 of 7 — Exponential Growth and Decay

The Model

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

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

• $k > 0$: exponential growth
• $k < 0$: exponential decay
• $y_0$: initial value

Half-Life and Doubling Time

Doubling time ($k > 0$): $T = \frac{\ln 2}{k}$

Half-life ($k < 0$): $T = \frac{\ln 2}{|k|}$

Exponential Models 🎯

Key Takeaways — Part 4

1. $\frac{dy}{dt} = ky$ has solution $y = y_0 e^{kt}$
2. Growth ($k>0$): doubling time $= \ln 2/k$
3. Decay ($k<0$): half-life $= \ln 2/|k|$

Differential Equations

Part 5 of 7 — More Separation of Variables Practice

Harder Examples

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

$\frac{dy}{y^2} = \frac{dx}{x}$

$-\frac{1}{y} = \ln|x| + C$

$y(1) = 2$: $-\frac{1}{2} = 0 + C$, so $C = -\frac{1}{2}$

$-\frac{1}{y} = \ln x - \frac{1}{2}$

$y = \frac{1}{\frac{1}{2} - \ln x} = \frac{2}{1 - 2\ln x}$

Separation of Variables Practice 🎯

Key Takeaways — Part 5

Practice various types of separable DEs to build fluency.

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! ✅