Order and Linearity

A precise classification guide for order, linearity, homogeneity, autonomy, and systems of differential equations.

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

The order of a differential equation is the highest derivative of the unknown function that appears. An $n$th-order ODE is linear if it can be arranged as $a_n(x)y^{(n)}+\cdots+a_1(x)y'+a_0(x)y=g(x)$, where coefficients and forcing depend only on the independent variable and the unknown function and derivatives appear to first power without products.

Notation and mathematical language

A linear equation is homogeneous when $g(x)=0$ and nonhomogeneous when $g\ne0$. This differs from the first-order scaling term 'homogeneous equation.' An equation is autonomous when the independent variable does not appear explicitly. Degree is not generally defined when derivatives occur inside non-polynomial expressions.

Conceptual picture

Classification predicts available structure. Linearity permits superposition for the homogeneous problem; order predicts the dimension of a solution space under regular coefficients and the typical number of initial values. Autonomy enables phase-line or phase-plane analysis.

Fully worked example

Interpretation and application

Classification is a decision tool, not a solution. A nonlinear first-order equation can be easier than a high-order linear one, and variable coefficients may block elementary formulas. Still, the labels prevent applying superposition or characteristic roots where they are invalid.