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Separable Equations (AP Calculus AB Rundown)

Introduction

Separable equations are a specific type of differential equation that can be solved using a relatively straightforward method. They are frequently encountered in AP Calculus AB and represent a powerful tool for modeling various phenomena. A differential equation is separable if it can be written in a form where the variables can be “separated” to opposite sides of the equation.

General Form

A first-order differential equation is considered separable if it can be written in the form:

$$ \frac{dy}{dx} = f(x)g(y) $$
where $ f(x) $ is a function of $ x $ only, and $ g(y) $ is a function of $ y $ only.

Solution Method

1. Separate the variables: Rewrite the equation so that all terms involving $ y $ (including $ dy $ ) are on one side and all terms involving $ x $ (including $ dx $ ) are on the other side. This typically involves algebraic manipulation.

$$ \frac{1}{g(y)} dy = f(x) dx $$
Division by Zero

2. Integrate both sides: Apply the integral to both sides of the equation with respect to their respective variables.

$$ \int \frac{1}{g(y)} dy = \int f(x) dx $$
Integration Techniques

3. Solve for $ y $ (if possible): After integrating, you will have an equation involving $ x $ and $ y $ . Attempt to isolate $ y $ to express the solution explicitly as $ y = F(x) $ . If this isn’t feasible, leave the solution in implicit form.

Implicit Differentiation

4. Apply initial conditions (if given): If an initial condition is provided (e.g., $ y(x_0) = y_0 $ ), substitute the values of $ x_0 $ and $ y_0 $ into the solution to determine the constant of integration ( $ C $ ) from the integration step.

Initial Value Problems

Example

Solve the differential equation:

$$ \frac{dy}{dx} = xy^2 $$

1. Separate Variables: $$ \frac{1}{y^2} dy = x dx $$
2. Integrate Both Sides: $$ \int \frac{1}{y^2} dy = \int x dx $$ $$ -\frac{1}{y} = \frac{x^2}{2} + C $$
3. Solve for $ y $ : $$ y = -\frac{1}{\frac{x^2}{2} + C} = -\frac{2}{x^2 + 2C} $$ We can replace $ 2C $ with a new constant, say $ K $ . $$ y = -\frac{2}{x^2 + K} $$

Common Applications

Separable equations are used in a wide variety of applications, including:

• Exponential Growth and Decay: Modeling population growth, radioactive decay, and other processes where the rate of change is proportional to the current quantity. Exponential Growth and Decay
• Newton’s Law of Cooling: Describing the cooling or heating of an object in a surrounding medium. Newton’s Law of Cooling
• Mixing Problems: Analyzing the concentration of substances in tanks or containers with inflow and outflow. Mixing Problems

This rundown provides a fundamental understanding of separable equations. Practice solving various examples to solidify your understanding and prepare for the AP Calculus AB exam.