Separating variables provides a structured method for solving differential equations by isolating dependent and independent variable terms, allowing students to integrate each side and derive an implicit general solution.

Separating Variables in First-Order Differential Equations

Separable differential equations form one of the most important families of equations students encounter in AP Calculus AB, because their structure allows direct manipulation into an integrable form. A differential equation is considered separable when its expression for the derivative can be rewritten so that all terms involving the dependent variable appear on one side and all terms involving the independent variable appear on the other. This process leads to a pathway for integrating both sides, ultimately obtaining a general solution containing an arbitrary constant.

When using separation of variables, it is essential to remember that the differential $dy$ and $dx$ should be treated symbolically as part of the notation rather than literal quantities. The method works because the derivative $dy/dx$ represents a ratio of infinitesimal changes, allowing algebraic manipulation under appropriate conditions.

A graph of a parabola illustrates how $dx$ is a small horizontal change and $dy$ is the corresponding vertical change, reinforcing the interpretation of $\dfrac{dy}{dx}$ as a ratio of infinitesimal changes. The figure visually supports the symbolic manipulation of differentials used in separation of variables. No additional mathematical content beyond the notes is included. Source.

Structure of a Separable Differential Equation

A separable differential equation typically begins in the form $dy/dx = f(x)g(y)$, where the rate of change of the dependent variable can be expressed as a product of a function of $x$ and a function of $y$. Once recognized, the goal is to isolate the expressions involving $y$ and $dy$ from those involving $x$ and $dx$.

After recognizing separability, students rewrite the equation by dividing or multiplying to group both $y$-related terms with $dy$ and both $x$-related terms with $dx$. This prepares the equation for the antidifferentiation step.

A typical rearrangement follows this structure:
$g(y)^{-1} dy = f(x) dx$.
This form ensures that each variable appears only on one side of the equation, creating a clear path to integration.

Integrating Both Sides

Once separation is complete, the next step is to antidifferentiate each side with respect to its variable. This step produces an implicit equation, because the antiderivatives may not naturally solve for the dependent variable.

The result of this process is an implicit general solution, which must include a constant of integration to account for the infinitely many solutions satisfying the original differential equation.

Between the integration steps, it is important for students to understand that both sides of the equation represent antiderivatives, and so each side individually would produce its own constant. However, when solving equations of this form, these constants can be combined into a single overall constant added on one side.

Steps in Solving by Separation of Variables

To solve a separable differential equation effectively, students can follow a systematic sequence of steps. This process ensures correct manipulation and integration.

Step 1: Identify separability

The equation must be expressible so that $dy/dx$ equals a product of a function of $x$ and a function of $y$. Students should check that all $y$-related terms can be grouped and all $x$-related terms can be grouped.

Step 2: Rearrange to isolate variables

Students place all $y$ values and $dy$ on one side, and all $x$ values and $dx$ on the other.

A side-by-side illustration contrasts mixed-variable differential equations with their separated counterparts, highlighting the rearrangement needed so that $x$-dependent and $y$-dependent expressions lie on opposite sides. The image aligns directly with the procedural step described in the notes, though it includes additional example equations that follow the same principle. Source.

Step 3: Integrate both sides

Using standard integration techniques, students antidifferentiate each side with respect to its associated variable. Typical integrals may include power functions, exponential functions, trigonometric functions, or logarithmic expressions depending on the form of the equation.

Step 4: Include the constant of integration

The constant must be included because differential equations describe entire families of solutions. Whether left on the $x$ side or within the implicit expression, the constant ensures that all valid solutions are represented.

Step 5: (When possible) Solve for the dependent variable

Though not always required to complete the solution, writing the solution explicitly for $y$ can make interpretation easier. Some differential equations, especially those involving logarithmic or exponential relationships, allow relatively straightforward solving for $y$. In cases where isolating $y$ is overly complex, the implicit form remains a valid and complete solution.

Importance of the Implicit General Solution

The final implicit solution produced through separation of variables communicates the relationship between the variables without necessarily providing an explicit function. This is acceptable and expected in many cases. Because implicit solutions still satisfy the original differential equation, they serve as the foundation for determining particular solutions once initial conditions are introduced in later subsubtopics.

Through careful manipulation, integration, and interpretation, students see how separable differential equations form a bridge between rate-based models and the functions that satisfy them, reinforcing core calculus ideas about change and accumulation.