First-Order Differential Equations

First-Order ODEs

A first-order differential equation involves only first derivatives and has the general form:

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

Separable equations occur when $f(x,y) = g(x)h(y)$. Rearrange and integrate both sides:

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

Linear first-order equations match the standard form $y' + P(x)y = Q(x)$. Solve them using the integrating factor $\mu$:

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

Worked Example: Radioactive decay is modeled by $\frac{dN}{dt} = -\lambda N$. Separating variables yields the solution $N = N_0 e^{-\lambda t}$, showing exponential decrease over time.

Core Types & Pitfalls

First-order equations fall into four main categories, each requiring a specific solution strategy:

Physics Link: Newton's law of cooling, $\frac{dT}{dt} = -k(T - T_{\text{env}})$, is a linear first-order ODE governing thermal equilibrium.

Common Pitfall: The general solution of a first-order ODE requires an arbitrary constant. Forgetting +C (or ignoring initial conditions) yields a specific solution rather than the complete solution required for your problem.