Unit 7: Differential Equations - Formulation and Visualizations

Modeling with Differential Equations

In AP Calculus AB, a Differential Equation (DE) is simply an equation containing one or more derivatives of an unknown function. Before you solve these equations, you must learn how to translate real-world physical descriptions into mathematical statements involving derivatives.

Translating Words to Math

The key to modeling is recognizing that a "rate of change" corresponds to a derivative. If $y$ represents a quantity, $\frac{dy}{dt}$ represents how fast that quantity changes with respect to time.

Here represents a translation guide for common phrases found in AP Exam free-response questions:

Key Variables:

• $k$: The constant of proportionality. If growth is increasing, $k > 0$. If obtaining decay, $k < 0$.
• $t$: The independent variable (usually time).
• $y$ (or $P$, $Q$, etc.): The dependent variable.

Example: Formulation

Scenario: A bacteria culture $B$ grows at a rate proportional to the square root of the amount of bacteria present.

Model:
Let $B(t)$ be the amount of bacteria at time $t$. The "rate" is $\frac{dB}{dt}$.
\frac{dB}{dt} = k\sqrt{B}

Verifying Solutions for Differential Equations

Unlike algebraic equations where the solution is a number (e.g., $x=5$), the solution to a differential equation is a function (e.g., $y = e^x$). You do not always need to solve a DE from scratch to verify if a function is a valid solution.

The Verification Process

To determine if a given function $y = f(x)$ is a solution to a differential equation:

1. Differentiate the proposed solution to find $y'$ (and $y''$ if the DE is second-order).
2. Substitute the function $y$ and its derivative $y'$ into the differential equation.
3. Simplify both sides. If the Left Hand Side (LHS) equals the Right Hand Side (RHS), the function is a solution.

General vs. Particular Solutions

• General Solution: Contains an arbitrary constant $+C$ (e.g., $y = x^2 + C$). It represents a family of functions.
• Particular Solution: A specific function where $C$ has been solved using an Initial Condition (e.g., $y(0) = 3$).

Worked Example: Verification

Problem: Verify that $y = e^{-2x}$ is a solution to the differential equation $y' + 2y = 0$.

Step 1: Find the derivative.
y = e^{-2x}
y' = -2e^{-2x}

Step 2: Substitute into the DE.
(-2e^{-2x}) + 2(e^{-2x}) = 0

Step 3: Simplify.
-2e^{-2x} + 2e^{-2x} = 0
0 = 0

Since the equation holds true, $y = e^{-2x}$ is indeed a solution.

Slope Fields

A Slope Field (or Direction Field) is a graphical tool used to visualize the general solution to a first-order differential equation. Since $\frac{dy}{dx}$ represents the slope of the tangent line, a slope field plots these slopes at various points on a coordinate grid.

Constructing a Slope Field

To sketch a slope field for $\frac{dy}{dx} = f(x,y)$:

1. Create a grid of points (e.g., integer coordinates like $(0,0), (1,1)$, etc.).
2. At each point $(x,y)$, calculate the value of the derivative $\frac{dy}{dx}$.
3. Draw a short line segment at that point with the calculated slope.

Interpreting Slope Fields (The "Flow")

Think of a slope field as a map of wind currents or water flow. A Solution Curve must follow the path of the segments. It must be differentiable (smooth) and cannot cross undefined regions.

• If you are given an initial condition $(x0, y0)$, start at that point and "follow the flow" to sketch the particular solution curve.
• Solution curves in a slope field never cross each other (due to the Uniqueness Theorem).

Patterns in Slope Fields

Recognizing patterns helps you match equations to graphs on multiple-choice questions:

1. Dependency on $x$ only: If $\frac{dy}{dx} = f(x)$ (e.g., $\frac{dy}{dx} = 2x$), the slopes are identical within every column. Vertical translation does not change the slope.
2. Dependency on $y$ only: If $\frac{dy}{dx} = f(y)$ (e.g., $\frac{dy}{dx} = y - 1$), the slopes are identical within every row. Horizontal translation does not change the slope.
3. Equilibrium Solutions: If for some constant $c$, $\frac{dy}{dx} = 0$ when $y=c$, look for a row of horizontal segments along the line $y=c$. This is a horizontal asymptote for the solution curves.

Common Mistakes & Pitfalls

• Slope Calculation Errors: Students often mentally swap $x$ and $y$ when plugging into the differential equation (e.g., for $\frac{dy}{dx} = y/x$, calculating $x/y$ instead). Tip: Write out a mini-table of coordinate pairs before drawing.
• Undefined Slopes: If the derivative is undefined (e.g., division by zero), do not draw a vertical line segment. Leave that point blank on the slope field grid.
• Ignoring Solution Domain: When sketching a solution curve through a slope field, you cannot cross vertical asymptotes. Solution curves are continuous functions defined on an interval.