Differential Equations

Differential Equations

Part 1 of 7 — Introduction to Differential Equations

Table of Contents

1. What is a Differential Equation?
2. Separation of Variables
3. Slope Fields
4. Exponential Growth & Decay
5. Particular Solutions & IVPs
6. Problem-Solving Workshop
7. Comprehensive Assessment

What is a Differential Equation?

A differential equation (DE) is an equation that relates a function to one or more of its derivatives.

Key Fact: On the AP Calculus AB exam, you only need to solve separable DEs and those solvable by direct integration.

Solving by Direct Integration

When $\frac{dy}{dx} = f(x)$ (right side depends only on $x$):

$\boxed{y = \int f(x)\,dx + C}$

Step-by-Step Process

General vs. Particular Solutions

AP Tip: ALWAYS write "$+C$" when finding a general solution. Forgetting $+C$ is one of the most common point-losing mistakes on FRQs.

Direct Integration Practice 🎯

Double Integration (Second-Order Direct)

When given $\frac{d^2y}{dx^2} = f(x)$ with two conditions:

$\boxed{\text{Integrate twice, using one condition each time}}$

Example: $f''(x) = 12x$, $f'(0) = -2$, $f(0) = 5$.

Step 1: $f'(x) = \int 12x\,dx = 6x^2 + C_1$

$f'(0) = C_1 = -2$, so $f'(x) = 6x^2 - 2$

Step 2: $f(x) = \int (6x^2 - 2)\,dx = 2x^3 - 2x + C_2$

$f(0) = C_2 = 5$, so $f(x) = 2x^3 - 2x + 5$

Key Fact: Each integration introduces one constant, each initial condition determines one constant.

Classify each DE. 🔍

Solve an IVP. ✍️

Key Takeaways — Part 1

Up Next: Part 2 — Separation of Variables.

Differential Equations

Part 2 of 7 — Separation of Variables

The Most Important DE Technique on AP AB

A DE is separable if it can be written as:

$\boxed{\frac{dy}{dx} = f(x) \cdot g(y)}$

The 5-Step Method

Key Fact: You only need ONE constant $C$ (not $C_1$ and $C_2$ on each side). The constants combine.

Common Separable Patterns

Worked Example — Careful with Signs

$\frac{dy}{dx} = -\frac{x}{y}$, $y(3) = 4$

$y\,dy = -x\,dx$

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

$y^2 = -x^2 + 2C$, i.e., $x^2 + y^2 = K$

$y(3) = 4$: $9 + 16 = K = 25$

$x^2 + y^2 = 25 \quad \text{(a circle!)}$

Separation of Variables 🎯

Common Mistakes in Separation

Domain Restrictions

$\boxed{\text{When dividing by } g(y), \text{ check if } g(y) = 0 \text{ gives an equilibrium solution}}$

Example: $\frac{dy}{dx} = y(3-y)$

Dividing by $y(3-y)$ loses $y = 0$ and $y = 3$ — these ARE constant solutions (equilibria).

Analyze each DE. 🔍

Solve a separable IVP. ✍️

Key Takeaways — Part 2

Up Next: Part 3 — Slope Fields.

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 $(x, y)$, draw a short line segment with slope $\frac{dy}{dx}$.

$\boxed{\text{Slope field shows } \frac{dy}{dx} \text{ at every point — no solving needed!}}$

Reading Slope Fields — Key Patterns

Key Fact: Solution curves follow the slope field like a river current. A solution through any point must be tangent to the slope segments.

Matching DEs to Slope Fields

Strategy: Check slopes at specific test points.

Isoclines

An isocline is a curve where all slopes are equal.

For $\frac{dy}{dx} = x + y$, the isocline where slope $= k$ is the line $x + y = k$ (or $y = k - x$).

AP Tip: On the AP exam, you might be asked to sketch a solution curve through a given point on a slope field. Follow the arrows smoothly — don't make sharp corners!

Slope Field Analysis 🎯

Sketching Solution Curves

Rules for sketching on a slope field:

Stability of Equilibria

For an autonomous DE $\frac{dy}{dx} = g(y)$:

Example: $\frac{dy}{dx} = y(2-y)$

• $y = 0$: unstable (slopes point away for $y < 0$ and toward for $y > 0$... actually for $y < 0$, $g(y) = y(2-y) < 0$ pushes $y$ more negative → unstable)
• $y = 2$: stable (solutions above and below approach $y = 2$)

Analyze slope fields. 🔍

Point analysis. ✍️

Key Takeaways — Part 3

Up Next: Part 4 — Exponential Growth & Decay.

Differential Equations

Part 4 of 7 — Exponential Growth & Decay

The Fundamental Model

$\boxed{\frac{dy}{dt} = ky \implies y = y_0 e^{kt}}$

Deriving the Solution

$\frac{dy}{y} = k\,dt$ → $\ln|y| = kt + C$ → $y = e^{kt+C} = e^C \cdot e^{kt} = Ae^{kt}$

At $t = 0$: $y(0) = A = y_0$. So $y = y_0 e^{kt}$.

Key Fact: "$y$ changes at a rate proportional to $y$" is the verbal form of $\frac{dy}{dt} = ky$.

Doubling Time & Half-Life

Quick Computation Tricks

Example: Substance has half-life 10 hours. Starting with 200 g:

• After 10 hrs: $100$ g
• After 20 hrs: $50$ g
• After 30 hrs: $25$ g

Finding $k$: $T_h = \ln 2 / |k|$ → $k = -\ln 2 / 10 \approx -0.0693$

Exponential Models 🎯

Newton's Law of Cooling

$\boxed{\frac{dT}{dt} = -k(T - T_{env})}$

Solution: Let $u = T - T_{env}$, then $\frac{du}{dt} = -ku$:

$T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$

Example: Coffee at $200°F$ in a $70°F$ room, $k = 0.1$:

$T(t) = 70 + 130e^{-0.1t}$

As $t \to \infty$: $T \to 70°F$ (room temperature).

AP Tip: Newton's Law of Cooling is a VERY common AP FRQ topic. The key insight: the rate of cooling is proportional to the temperature DIFFERENCE, not the temperature itself.

Classify exponential models. 🔍

Apply Newton's Law of Cooling. ✍️

Key Takeaways — Part 4

Up Next: Part 5 — Particular Solutions & IVPs.

Differential Equations

Part 5 of 7 — Particular Solutions & Advanced IVPs

Harder Separable DEs

Not all separable DEs give $y = Ae^{kt}$. Here are the main solution forms:

Worked Example 1: $y^2$ Type

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

$\frac{dy}{y^2} = \frac{dx}{x}$ → $-\frac{1}{y} = \ln|x| + C$

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

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

Worked Example 2: Square Root Type

$\frac{dy}{dx} = \frac{x}{\sqrt{y}}, \quad y(0) = 4$

$\sqrt{y}\,dy = x\,dx$ → $\frac{2}{3}y^{3/2} = \frac{x^2}{2} + C$

$y(0) = 4$: $\frac{2}{3}(8) = C = \frac{16}{3}$

$y^{3/2} = \frac{3x^2}{4} + 8$

AP Tip: On AP FRQs, it's acceptable to leave the answer in implicit form (not solved for $y$) unless the problem specifically says "solve for $y$".

Advanced Separation 🎯

Domain of Solutions

$\boxed{\text{The domain of a particular solution may be restricted!}}$

When finding a particular solution, always check:

Key Fact: A particular solution exists on the largest interval containing the initial point where the solution is continuous and the DE is defined.

Analyze solutions. 🔍

Find a particular solution value. ✍️

Key Takeaways — Part 5

Up Next: Part 6 — Problem-Solving Workshop.

Differential Equations

Part 6 of 7 — Problem-Solving Workshop

AP FRQ Strategy Guide

Euler's Method

$\boxed{y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x}$

Example: $\frac{dy}{dx} = x + y$, $y(0) = 1$, $\Delta x = 0.5$. Approximate $y(1)$.

So $y(1) \approx 2.5$.

Euler's Method Accuracy

AP Tip: Euler's method questions typically ask for 2-3 steps. Set up a TABLE — it's the clearest way to show work.

AP-Style Workshop 🎯

Common AP FRQ Patterns

Pattern 1: Rate In − Rate Out

A tank has water flowing in at rate $R_{in}$ and out at rate $R_{out}$:

$\frac{dV}{dt} = R_{in} - R_{out}$

Pattern 2: Logistic Growth (BC topic, but concept appears in AB)

$\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$

• $P < L$: population grows
• $P > L$: population decreases
• $P = L$: equilibrium (carrying capacity)
• Fastest growth at $P = L/2$

Pattern 3: Using $\frac{dy}{dx}$ from a Table

Given a table of $(x, y, dy/dx)$ values, use Euler's method or verify a proposed solution.

Classify and solve. 🔍

Euler's method computation. ✍️

Key Takeaways — Part 6

Up Next: Part 7 — Comprehensive Assessment.

Differential Equations

Part 7 of 7 — Comprehensive Assessment

Complete Reference

Top AP Mistakes

Final Assessment — Set 1 🎯

Final Assessment — Set 2 🎯

Final classification. 🔍

Final challenge. ✍️

Differential Equations — Complete! ✅

Congratulations! You've mastered differential equations for AP Calculus AB.