Differential models in one variable

Differential Models & Separation

A differential equation (DE) relates an unknown function to its derivatives, describing how systems evolve in time or space.

Separable DEs are equations where algebraic manipulation isolates the variables onto opposite sides:

$$\frac{dy}{dx} = g(x)h(y) \implies \int \frac{1}{h(y)} dy = \int g(x) dx$$

Follow this Step-by-step method to solve them:

Initial conditions supply a known state like $y(t_0)=y_0$ to fix the integration constant $C$ uniquely.

Classic Models & Pitfalls

Exponential growth / decay models systems where change is proportional to current size: $\frac{dN}{dt} = kN$, yielding $N(t) = N_0 e^{kt}$.

Newton's law of cooling states rate of temperature change is proportional to the difference from the environment: $\frac{dT}{dt} = -\lambda(T - T_{\text{env}})$. Its solution is $T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})e^{-\lambda t}$.

Common pitfall: Do not confuse rate phrasing. "Grows at a rate proportional to itself" means $\frac{dN}{dt} = kN$ ($e^{kt}$), whereas "grows at a constant rate" means a linear slope. Always check unit consistency for dimensional safety.