Quick questions on First-order differential equations and separation of variables (QCE Specialist Mathematics Unit 4)

If a first-order equation can be written as $\dfrac{dy}{dx} = f(x)g(y)$, separate the variables so each side involves only one variable, then integrate both sides:

The rate of cooling is proportional to the temperature difference from the surroundings $T_s$:

When growth is limited by a carrying capacity $M$, the model is

Always verify a solution by substituting it back into the original equation. Differentiate your $y(t)$ and confirm it reproduces the stated rate, and check that the initial condition is satisfied. A second sanity check is the long-term behaviour: a decay model should tend to zero, a cooling model should tend to the ambient temperature, and a logistic model should tend to the carrying capacity. If the long-run behaviour contradicts the physical setup, a sign error in $k$ or a dropped constant is the usual cause.