🎯⭐ INTERACTIVE LESSON

Differential Equations

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Differential Equations - Complete Interactive Lesson

Part 1: Introduction to Differential Equations

Differential Equations

Part 1 of 7 — Introduction to Differential Equations

What is a Differential Equation?
Separation of Variables
Slope Fields
Exponential Growth & Decay
Particular Solutions & IVPs
Problem-Solving Workshop
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.

Type	Example	Method
Directly integrable	$\frac{dy}{dx} = 3x^2$	Integrate both sides
Separable	$\frac{dy}{dx} = 2y$	Separate and integrate
Non-separable	$\frac{dy}{dx} = x + y$	Not on AP AB

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 $x$ ):

Direct Integration Practice 🎯

Double Integration (Second-Order Direct)

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

Classify each DE. 🔍

Solve an IVP. ✍️

Key Takeaways — Part 1

Concept	Key Point
Differential equation	Relates a function to its derivatives
Direct integration	When $dy/dx = f(x)$ — integrate both sides
General solution	Includes $+C$ (family of curves)
Particular solution	determined by initial condition

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:

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

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 $\frac{dy}{dx}$ .

Part 4: Exponential Growth and Decay

Differential Equations

Part 4 of 7 — Exponential Growth & Decay

The Fundamental Model

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

Part 5: More Separation of Variables Practice

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:

DE Type	Separation	Solution Form
$\frac{dy}{dx} = ky$

Part 6: AP-Style Workshop

Differential Equations

Part 6 of 7 — Problem-Solving Workshop

AP FRQ Strategy Guide

Problem Type	What to Do
"Find the particular solution"	Separate, integrate, apply IC, solve for $y$
"Sketch solution on slope field"	Follow slope segments smoothly from given point
"Find equilibrium solutions"	Set $dy/dx = 0$ , solve for $y$

Part 7: Comprehensive Assessment

Differential Equations

Part 7 of 7 — Comprehensive Assessment

Complete Reference

Topic	Key Formula
Direct integration	$dy/dx = f(x) \implies y = \int f(x)\,dx + C$

y = \int f (x) d x + C

Step-by-Step Process

Step	Action	Example: $\frac{dy}{dx} = 6x^2 - 4x + 1$ , $y(0) = 3$
1	Integrate both sides	$y = \int (6x^2 - 4x + 1)\,dx$
2	Find antiderivative

General vs. Particular Solutions

Term	Meaning	Example
General solution	Family of curves (includes $C$ )	$y = 2x^3 - 2x^2 + x + C$
Particular solution	One specific curve ( $C$ determined)	$y = 2x^3 - 2x^2 + x + 3$
Initial condition	Point that determines $C$	$y(0) = 3$

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

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

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.

Up Next: Part 2 — Separation of Variables.

Step	Action	Example: $\frac{dy}{dx} = xy$ , $y(0) = 2$
1	Separate variables	$\frac{dy}{y} = x\,dx$
2	Integrate both sides	$\int \frac{dy}{y} = \int x\,dx$

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

Common Separable Patterns

DE Form	Separation	General Solution
$\frac{dy}{dx} = ky$	$\frac{dy}{y} = k\,dx$	$y = Ae^{kx}$
$\frac{dy}{dx} = \frac{x}{y}$
$\frac{dy}{dx} = xy^2$	$\frac{d y}{y^{2}} = x$
$\frac{dy}{dx} = \frac{y}{x}$
$\frac{dy}{dx} = y(1-y)$	$\frac{d y}{y (}$

Worked Example — Careful with Signs

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

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

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

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

Mistake	Problem	Correction
Forgetting to separate ALL $y$ 's	Leaving $y$ on the $dx$ side	Move everything with $y$ to one side
Dividing by $g(y) = 0$	Lose equilibrium solutions!	Check: does $g(y) = 0$ give solutions?
Wrong sign on $\ln	y	$
Forgetting absolute value	$\ln y$ vs $\ln	y
Two constants of integration	$C_1$ on left AND $C_2$ on right	Combine into single $C$ on one side

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

Concept	Key Rule
Separable DE	$dy/dx = f(x) \cdot g(y)$
Method	Separate, integrate, solve, apply IC
Only one $C$	Constants combine — use $C$ on one side only
Equilibrium solutions	Set $g(y) = 0$ — don't lose them!
$\int dy/y = \ln	y

Up Next: Part 3 — Slope Fields.

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

Reading Slope Fields — Key Patterns

Observation	What It Tells You
All slopes horizontal along $y = c$	$\frac{dy}{dx} = 0$ when $y = c$ (equilibrium)
Slopes same along horizontal lines	DE depends only on $y$ : $\frac{dy}{dx} = g(y)$
Slopes same along vertical lines	DE depends only on $x$ : $\frac{dy}{dx} = f(x)$
Slopes same along diagonal $y = x + c$	DE depends on $y - x$
Slopes get steeper as $	y

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.

Test Point	$\frac{dy}{dx} = x$	$\frac{dy}{dx} = y$	$\frac{dy}{dx} = x+y$	$\frac{dy}{dx} = xy$
$(0,0)$	$0$	$0$	$0$	$0$

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 (or ).

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:

Rule	Why
Solution must be tangent to every segment it crosses	By definition of slope
Solutions cannot cross each other	Uniqueness theorem (unless DE is undefined)
Curve must be smooth (no corners)	Solutions to these DEs are differentiable
Follow the "flow" of the field	Solutions are carried along like water

Stability of Equilibria

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

Equilibrium Type	Slope Field Behavior	Stability
Stable	Arrows point TOWARD equilibrium	Solutions approach it
Unstable	Arrows point AWAY from equilibrium	Solutions diverge
Semi-stable	Arrows point toward on one side, away on other	Mixed behavior

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

$y = 0$ : unstable (slopes point away for $y < 0$ and toward for $y > 0$ ... actually for $y < 0$ , pushes more negative → )

Analyze slope fields. 🔍

Point analysis. ✍️

Key Takeaways — Part 3

Concept	Key Point
Slope field	Visual: short segments showing $dy/dx$ at each point
Matching DEs	Evaluate $dy/dx$ at test points
Solution curves	Must be tangent to segments, smooth, non-crossing
Equilibrium	Where $dy/dx = 0$
Stable/Unstable	Do nearby solutions approach or diverge?

Up Next: Part 4 — Exponential Growth & Decay.

Parameter	Meaning
$y_0$	Initial value: $y(0) = y_0$
$k > 0$	Exponential growth
$k < 0$	Exponential decay
$t$	Time variable

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

Concept	Formula	Derivation
Doubling time $(k > 0)$	$T_d = \frac{\ln 2}{k}$	$2y_0 = y_0 e^{kT}$ , $\ln$
Half-life $(k < 0)$	$T_h = \frac{\ln 2}{	k

Quick Computation Tricks

Number of half-lives	Fraction remaining
$1$	$1/2$
$2$	$1/4$
$3$

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

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

Variable	Meaning
$T$	Temperature of object
$T_{env}$	Ambient (surrounding) temperature
$k > 0$

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

Model	DE	Solution
Exponential growth	$dy/dt = ky$ , $k>0$	$y = y_0 e^{kt}$
Exponential decay	$dy/dt = ky$ , $k<0$	$y = y_0 e^{kt}$
Newton's cooling	$dT/dt = -k(T-T_{env})$	$T = T_{}$
Doubling time	—	$T_d = \ln 2/k$
Half-life	—	$T_h = \ln 2/

Up Next: Part 5 — Particular Solutions & IVPs.

Worked Example 1: $y^2$ Type

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

$\frac{dy}{y^2} = \frac{dx}{x}$ →

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

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

Worked Example 2: Square Root Type

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

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

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

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

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:

Issue	Example	Domain Restriction
Division by zero	$y = \frac{2}{1-2\ln x}$	$x \neq e^{1/2}$
Square root of negative	$y = \sqrt{9-x^2}$
Logarithm of non-positive	$\ln	y
Continuity through IC	Must connect to initial point	Choose interval containing IC

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

Concept	Key Point
$1/y^2$ DEs	Give $-1/y = F(x) + C$ solutions
$\sqrt{y}$ DEs	Give $y^{3/2} = F(x) + C$ solutions
Domain restrictions	Check for division by zero, square roots
IC determines branch	Positive IC → positive root
Implicit solutions OK	Don't need to solve for $y$ unless asked

Up Next: Part 6 — Problem-Solving Workshop.

Euler's Method

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

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

Step	$x_n$	$y_n$	$f (x_{n}, y_{n})$

So $y(1) \approx 2.5$ .

Euler's Method Accuracy

Condition	Euler's result is...
Solution is concave up	Underestimate (tangent line below curve)
Solution is concave down	Overestimate (tangent line above curve)
Smaller $\Delta x$	More accurate

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

Topic	Key Formula/Idea
Euler's method	$y_{n+1} = y_n + f(x_n,y_n) \cdot \Delta x$
Concave up → Euler	Underestimate
Concave down → Euler	Overestimate
Equilibrium	$dy/dx = 0$ : constant solutions
Newton's cooling	$dT/dt = -k(T - T_{env})$

Up Next: Part 7 — Comprehensive Assessment.

Mistake	Fix
Forgetting $+C$	ALWAYS include until IC is applied
Not separating completely	ALL $y$ 's on one side, ALL $x$ 's on the other
Losing equilibrium solutions	Check $g(y)=0$ before dividing
Wrong Euler direction	Concave up = underestimate
Forgetting domain	Check where solution is defined

Final Assessment — Set 1 🎯

Final Assessment — Set 2 🎯

Final classification. 🔍

Final challenge. ✍️

Differential Equations — Complete! ✅

Skill	Status
Direct integration	✅
Separation of variables	✅
Slope fields	✅
Exponential growth/decay	✅
Newton's cooling	✅
Euler's method	✅
Equilibrium & stability	✅
Particular solutions & domains	✅

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

y = 2 x^{3} - 2 x^{2} + x + C

\int \frac{d y}{y} = \int x d x

\frac{d y}{y ^{2}} = x d x

\frac{d y}{y ( 1 - y )} = d x

y = 2

: stable (solutions above and below approach

y = 2

)

T = T_{e n v} + (T_{0} - T_{e n v}) e^{- k t}

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

\frac{2}{3} y^{3/2} = \frac{x ^{2}}{2} + C

f (x_{n}, y_{n}) = x_{n} + y_{n}

Differential Equations

Step-by-Step Process

General vs. Particular Solutions

The 5-Step Method

Common Separable Patterns

Worked Example — Careful with Signs

Common Mistakes in Separation

Domain Restrictions

Key Takeaways — Part 2

Reading Slope Fields — Key Patterns

Matching DEs to Slope Fields

Isoclines

Sketching Solution Curves

Stability of Equilibria

Key Takeaways — Part 3

Deriving the Solution

Doubling Time & Half-Life

Quick Computation Tricks

Newton's Law of Cooling

Key Takeaways — Part 4

Worked Example 1: $y^2$ Type

Worked Example 2: Square Root Type

Domain of Solutions

Key Takeaways — Part 5

Euler's Method

Euler's Method Accuracy

Common AP FRQ Patterns

Key Takeaways — Part 6

Top AP Mistakes

Differential Equations — Complete! ✅

Differential Equations

Differential Equations - Complete Interactive Lesson

Part 1: Introduction to Differential Equations

Differential Equations

Table of Contents

What is a Differential Equation?

Solving by Direct Integration

Double Integration (Second-Order Direct)

Key Takeaways — Part 1

Part 2: Separation of Variables

Differential Equations

The Most Important DE Technique on AP AB

Part 3: Slope Fields

Differential Equations

What is a Slope Field?

Part 4: Exponential Growth and Decay

Differential Equations

The Fundamental Model

Part 5: More Separation of Variables Practice

Differential Equations

Harder Separable DEs

Part 6: AP-Style Workshop

Differential Equations

AP FRQ Strategy Guide

Part 7: Comprehensive Assessment

Differential Equations

Complete Reference

Step-by-Step Process

General vs. Particular Solutions

The 5-Step Method

Common Separable Patterns

Worked Example — Careful with Signs

Common Mistakes in Separation

Domain Restrictions

Key Takeaways — Part 2

Reading Slope Fields — Key Patterns

Matching DEs to Slope Fields

Isoclines

Sketching Solution Curves

Stability of Equilibria

Key Takeaways — Part 3

Deriving the Solution

Doubling Time & Half-Life

Quick Computation Tricks

Newton's Law of Cooling

Key Takeaways — Part 4

Worked Example 1: y2y^2y2 Type

Worked Example 2: Square Root Type

Domain of Solutions

Key Takeaways — Part 5

Euler's Method

Euler's Method Accuracy

Common AP FRQ Patterns

Key Takeaways — Part 6

Top AP Mistakes

Differential Equations — Complete! ✅

Worked Example 1: $y^2$ Type