Differential models in one variable

A differential equation (DE) relates a function to its derivatives. In physics, DEs describe how systems evolve.

Separable DEs

$$\frac{dy}{dx} = g(x)\,h(y) \;\;\Longrightarrow\;\; \int\frac{dy}{h(y)} = \int g(x)\,dx$$

Exponential growth / decay

$$\frac{dN}{dt} = kN \;\;\Longrightarrow\;\; N(t) = N_0\,e^{kt}$$

• $k>0$: population growth, compound interest.
• $k<0$: radioactive decay, cooling.

Newton's law of cooling

$$\frac{dT}{dt} = -\lambda\,(T - T_{\text{env}})$$

Solution: $T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})\,e^{-\lambda t}$.

Initial conditions — The constant $C$ from integration is fixed by a known state: $y(t_0)=y_0$.

Step-by-step method

1. Separate variables onto each side.
2. Integrate both sides.
3. Solve for $y$.
4. Apply the initial condition to find $C$.

Tip: Always check units. If $[k]=\text{s}^{-1}$, then $kt$ is dimensionless, which is consistent with the exponent of $e$.

Common pitfall: "Grows at a rate proportional to itself" gives $e^{kt}$ — but "grows at a constant rate" gives a straight line. Translating words to the wrong rate law dooms a model before any calculus happens.