First-Order Differential Equations

Modelling Change Over Time

A differential equation relates a function to its derivatives — it describes how something changes rather than what it is.

First-order ODEs involve the first derivative only: $\frac{dy}{dt} = f(t, y)$.

Separable equations: if $f(t,y) = g(t) h(y)$, separate variables and integrate both sides:

$$\frac{dy}{h(y)} = g(t)\, dt \qquad \Rightarrow \qquad \int \frac{dy}{h(y)} = \int g(t)\, dt$$

Linear first-order ODEs: $\frac{dy}{dt} + p(t) y = q(t)$. Solved by the integrating factor $\mu(t) = e^{\int p(t)\, dt}$:

$$y(t) = \frac{1}{\mu(t)} \int \mu(t) q(t)\, dt$$

The exponential growth/decay model: $\frac{dy}{dt} = ky$ has solution $y(t) = y_0 e^{kt}$. If $k > 0$: growth; if $k < 0$: decay. This is the simplest and most important ODE in economics.

$$y(t) = y_0 e^{kt}$$

Economic applications:

• Continuous compound interest: $\frac{dA}{dt} = rA$ → $A(t) = A_0 e^{rt}$.
• Price adjustment: $\frac{dp}{dt} = \alpha(D - S)$ — price rises when demand exceeds supply.
• Solow growth model: $\dot{k} = sf(k) - \delta k$ — capital accumulation as saving minus depreciation.
• Advertising decay: brand awareness $A(t) = A_0 e^{-\delta t}$ — without reinforcement, awareness fades exponentially.

The steady state (equilibrium) is where $dy/dt = 0$ — the system stops changing. Stability depends on whether deviations from equilibrium grow ($k > 0$: unstable) or shrink ($k < 0$: stable).

Stability criterion: for $\dot{y} = f(y)$ near equilibrium $y^*$: if $f'(y^*) < 0$, the equilibrium is stable (deviations shrink); if $f'(y^*) > 0$, it is unstable (deviations grow).

The first-order toolkit

Common pitfall: Losing solutions when separating. Dividing by $h(y)$ assumes $h(y) \ne 0$ — the constant solutions where $h(y) = 0$ (often the equilibria you care most about) must be checked separately.