First-Order Differential Equations

First-Order ODEs & Methods

A differential equation relates a function to its derivatives, describing how something changes. First-order ODEs involve only the first derivative: $\frac{dy}{dt} = f(t, y)$.

Separable equations occur when $f(t,y) = g(t) h(y)$. Separate variables and integrate:

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

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

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

Common pitfall: Dividing by $h(y)$ assumes $h(y) \neq 0$. Always check constant solutions where $h(y) = 0$ separately.

Economic Models & Stability

The exponential growth/decay model $\frac{dy}{dt} = ky$ yields $y(t) = y_0 e^{kt}$. If $k > 0$, it is growth; if $k < 0$, it is decay.

Key economic applications include:

• Compound interest: $\frac{dA}{dt} = rA \Rightarrow A_0 e^{rt}$
• Price adjustment: $\frac{dp}{dt} = \alpha(D - S)$
• Solow growth: $\dot{k} = sf(k) - \delta k$
• Advertising decay: $A(t) = A_0 e^{-\delta t}$

The steady state occurs where $dy/dt = 0$. For $\dot{y} = f(y)$ near equilibrium $y^*$, stability depends on the derivative sign.