Separable Equations

A separable equation is simply one where the right side splits into a part that only involves $t$ times a part that only involves $y$. The name says what that lets you do: separate the variables, with every $y$ on one side and every $t$ on the other. It covers a lot of real models, like cooling coffee, draining tanks and growing populations. All we're really saying with $\frac{dy}{dt} = g(t)\,h(y) \implies \int\frac{dy}{h(y)} = \int g(t)\,dt$ is that $g(t)$ (the $t$ part) and $h(y)$ (the $y$ part) can be pulled apart, and then each side is an ordinary one-variable integral. It looks complex, but it's just two integrals, so bear with it.

Take an equation like $y' = 6y^2t$. It isn't linear (there's a $y^2$), so integrating factors are out. But notice how the right side is built: a function of $t$ times a function of $y$. Equations of the form $y' = g(t)\,h(y)$ are called separable, and they're the next easiest kind after $y' = f(t)$.

Why separating works. Divide by $h(y)$ to get $\frac{1}{h(y)}\,y' = g(t)$. Now think about the left side. If $H$ is an antiderivative of $\frac1h$, the chain rule says $\frac{d}{dt}H(y(t)) = \frac{1}{h(y)}\,y'$. So the left side is a single derivative, $\frac{d}{dt}H(y)$, and integrating both sides in $t$ gives $H(y) = \int g(t)\,dt$. Writing $\frac{dy}{h(y)} = g(t)\,dt$ is just shorthand for that.

Steps:

1. Move every $y$ to the side with $dy$ and every $t$ to the side with $dt$.
2. Integrate both sides. One constant $C$ is enough, since two constants on opposite sides would just combine into one.
3. Use the initial condition (it's often easiest before solving for $y$).
4. Solve for $y$ explicitly if possible.

A common mistake. It's tempting to think separating finds every solution, because the steps never seem to throw anything away. They can. Dividing by $h(y)$ assumes $h(y) \ne 0$. Any value with $h(y) = 0$ gives a constant solution, and separation can miss it, so check those values separately.

Solutions to nonlinear equations can also blow up in finite time. The interval of validity is wherever the formula makes sense around the initial point; you'll see this in the example below, and we'll look at it properly a few lessons from now.

Exponential growth and decay, Newton's law of cooling, the logistic population model and the rate of a chemical reaction are all separable.