Introduction to Differential Equations

The Language of Change

A differential equation relates an unknown function to its derivatives, describing how quantities evolve in the physical world. For example, $\frac{dy}{dx} = 2y$ states that the rate of change of $y$ equals twice $y$.

A solution is a function that satisfies the equation. The order of a differential equation is the order of the highest derivative appearing in it.

Differential equations typically have a family of solutions containing a constant of integration. An initial condition (like $y(0) = 3$) pins down one particular solution.

Separable Equations

A first-order separable equation can be manipulated so that each variable is on its own side:

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

To solve, integrate both sides. For growth $\frac{dy}{dx} = ky$: separate to $\frac{dy}{y} = k\, dx$, integrate to $\ln|y| = kx + C$, and exponentiate to the exponential growth law $y = y_0 e^{kx}$.

This method solves radioactive decay, Newton's law of cooling $\frac{dT}{dt} = -k(T - T_{env})$, and population models.

Common Pitfall: Forgetting the constant of integration when integrating both sides. Omitting $+C$ treats the general solution as a single function and loses the infinite family of solutions.