First-Order Differential Equations

A first-order ODE has the general form:

$$\frac{dy}{dx} = f(x, y)$$

Separable equations — If $f(x,y) = g(x)\,h(y)$:

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

Linear first-order — Form $y' + P(x)y = Q(x)$:

$$y = \frac{1}{\mu}\int \mu\,Q\,dx, \quad \mu = e^{\int P\,dx}$$

Example — Radioactive decay: $dN/dt = -\lambda N \implies N = N_0 e^{-\lambda t}$

Common types

• Separable $y' = g(x)h(y)$: separate variables
• Linear $y' + Py = Q$: integrating factor
• Exact $M\,dx + N\,dy = 0$: find potential
• Bernoulli $y' + Py = Qy^n$: substitute $v = y^{1-n}$

Physics link: Newton's law of cooling $dT/dt = -k(T-T_{\text{env}})$ is a linear first-order ODE.

Common pitfall: The general solution of a first-order ODE carries one arbitrary constant — forgetting "+C" (or fixing it with the wrong initial condition) yields a solution but not your solution.