Introduction to Differential Equations

A rigorous orientation to differential equations, solution concepts, classification, modelling, and verification.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

A differential equation relates an unknown function to one or more derivatives. An ordinary differential equation uses derivatives with respect to one independent variable; a partial differential equation uses partial derivatives in several variables. A solution is a sufficiently differentiable function satisfying the equation at every point of a stated interval or region.

Notation and mathematical language

The order is the highest derivative present. A linear ODE has the unknown function and derivatives to first power with coefficients depending only on the independent variable. A general solution contains free constants; a particular solution fixes them, often through initial or boundary data.

Conceptual picture

A differential equation specifies local change rather than values directly. Many functions can share the same derivative rule until data select one. Exact symbolic formulas, implicit relations, qualitative phase portraits, and numerical tables are different valid representations with different information and error characteristics.

Fully worked example

Interpretation and application

Differential equations model motion, epidemics, circuits, fluids, finance, and feedback. Their predictions are conditional: parameter estimation, measurement error, omitted mechanisms, and model scope must be separated from errors in solving the equation itself.