4.3 Separable Equations

Separable equations are a type of differential equation where you can isolate the variables on opposite sides of the equation. They show up constantly in modeling real-world scenarios like cooling, population growth, and radioactive decay. Mastering the technique here gives you a reliable method you'll use throughout the rest of differential equations.

Separable Equations

Solving separable differential equations

A separable equation is any differential equation that can be written in the form $\frac{dy}{dx} = f(x)g(y)$, where the right-hand side is a product of a function of $x$ alone and a function of $y$ alone. That structure is what makes separation possible.

Here's the process for solving one:

1. Separate the variables. Divide both sides by $g(y)$ and multiply both sides by $dx$ so that all $y$-terms (including $dy$) are on one side and all $x$-terms (including $dx$) are on the other: $\frac{1}{g(y)}\,dy = f(x)\,dx$

2. Integrate both sides with respect to their own variable: $\int \frac{1}{g(y)}\,dy = \int f(x)\,dx$ These integrals may require techniques like $u$-substitution, partial fractions, trig identities, or logarithmic rules ($\int \frac{1}{x}\,dx = \ln|x| + C$).

3. Solve for $y$ as a function of $x$. This might mean exponentiating both sides if you end up with logarithms, or it might require some algebra. Don't forget the constant of integration, which appears as a single $+C$ on one side (you don't need one on both sides since they combine).

4. Apply the initial condition if one is given. Substitute the known point $(x_0, y_0)$ into your general solution and solve for $C$. This pins down the one particular solution that passes through that point.

In some cases you won't be able to isolate $y$ explicitly, and the solution stays in implicit form. That's fine; an implicit solution is still a valid answer.

Applications of separable equations

Solution concentrations. When the rate of change of a concentration is proportional to the current concentration, you get $\frac{dC}{dt} = kC$, where $k$ is a constant (positive for growth, negative for decay). Separating and integrating gives $C(t) = C_0 e^{kt}$, where $C_0$ is the initial concentration. You can use this to predict future concentrations or find how long it takes to reach a target value.

Newton's law of cooling. The rate of temperature change of an object is proportional to the difference between its temperature and the surrounding (ambient) temperature: $\frac{dT}{dt} = k(T - T_a)$

Here $T_a$ is the ambient temperature and $k$ is a negative constant. Solving gives $T(t) = T_a + (T_0 - T_a)e^{kt}$, which tells you the object's temperature at any time $t$. For example, if coffee at 90°C is placed in a 20°C room, this model predicts how quickly it cools.

Other common models using the same separable structure:

• Population growth: $\frac{dP}{dt} = kP$ models exponential growth (or decay if $k < 0$)
• Radioactive decay: $\frac{dN}{dt} = -\lambda N$, where the decay constant $\lambda$ determines the half-life
• Compound interest (continuous): $\frac{dA}{dt} = rA$, where $r$ is the interest rate and $A$ is the account value

Notice these all share the same form: the rate of change is proportional to the current amount. That's why the same separation technique solves all of them.

Analysis of separable equation solutions

Direction fields give you a visual picture of a differential equation's behavior without solving it. At each point $(x, y)$ in the plane, you evaluate $\frac{dy}{dx}$ and draw a short line segment with that slope. The collection of these segments shows you the general flow of solution curves, revealing trends, symmetry, and long-term behavior at a glance.

Initial conditions let you pick out one specific solution from the family of all solutions. The process:

1. Solve the separable equation to get the general solution (with a constant $C$).
2. Plug in the initial condition $(x_0, y_0)$ and solve for $C$.
3. Substitute that value of $C$ back into the general solution to get the particular solution.

Combining both tools is powerful. Sketch the direction field first to understand the overall landscape of solutions. Then plot your initial condition and trace the solution curve through it, following the slopes the field indicates. This confirms your algebraic solution and helps you catch errors.

Autonomous Equations and Equilibrium Solutions

An autonomous equation is a separable equation where the right-hand side depends only on $y$: $\frac{dy}{dx} = f(y)$. Because $x$ doesn't appear explicitly, the slope at any point depends only on the $y$-value. This means the direction field has identical slopes along every horizontal line.

Equilibrium solutions occur at values of $y$ where $f(y) = 0$. At these values, $\frac{dy}{dx} = 0$, so the solution is just a constant horizontal line $y = c$. These are the "resting states" of the system.

The phase line is a compact, one-dimensional tool for analyzing autonomous equations. You draw a vertical number line for $y$, mark the equilibrium points (where $f(y) = 0$), and then add arrows between them: upward where $f(y) > 0$ (solutions increase) and downward where $f(y) < 0$ (solutions decrease). This quickly tells you whether solutions move toward or away from each equilibrium, which is the key to understanding long-term behavior.