Section 9-3: Separable Equations

Welcome back to class! In this lesson, we dive into one of the most essential techniques in solving differential equations: Separable Equations. This method allows us to solve first-order differential equations where the variables can be factored and isolated on opposite sides of the equation.

The Core Concept

A separable equation is a first-order differential equation that can be written in the form:

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

The name "separable" comes from the fact that we can rewrite this equation so that all terms involving $y$ are on one side (with $dy$) and all terms involving $x$ are on the other side (with $dx$). The general strategy involves three main steps:

  1. Separate the variables: $\frac{1}{f(y)} dy = g(x) dx$
  2. Integrate both sides: $\int h(y) dy = \int g(x) dx$
  3. Solve for $y$ (explicitly if possible) or leave the solution in implicit form.

Example 1: Solving with Initial Conditions

Let's look at a classic example from the lecture notes. We are asked to solve the equation given the initial condition $y(0) = 2$:

$$\frac{dy}{dx} = \frac{x^2}{y^2}$$

First, we cross-multiply to separate the variables:

$$y^2 \, dy = x^2 \, dx$$

Next, we integrate both sides:

$$\int y^2 \, dy = \int x^2 \, dx \implies \frac{1}{3}y^3 = \frac{1}{3}x^3 + C$$

Using our initial condition $x=0$ and $y=2$, we can solve for $C$, finding that $C=8/3$ (or just adjusting the constant). The final specific solution becomes:

$$y = \sqrt[3]{x^3 + 8}$$

Real-World Application: Mixing Problems

Differential equations are incredibly useful for modeling physical reality. A key application covered in the notes is Mixing Problems (Example 6). Imagine a tank with salt water flowing in and out. The rate of change of salt in the tank is defined by:

$$\frac{dy}{dt} = (\text{Rate In}) - (\text{Rate Out})$$

In our class example, we modeled a tank containing $20$ kg of salt with brine flowing in. By setting up the differential equation $\frac{dy}{dt} = 0.75 - \frac{y}{200}$ and solving it, we determined that after 30 minutes, approximately $38.1$ kg of salt remains in the tank.

Class Resources

Please review the attached notes for more complex examples, including orthogonal trajectories and solutions involving exponential functions.

Keep practicing those integrals, and don't forget to apply the Chain Rule when checking your work!