So far, we’ve learned how to identify the core parts of a differential equation, its variables, terms, and coefficients. We now turn to how differential equations are classified according to the presence of these components.
In this section, we’ll focus on two of the most important classifications: the equation’s order and its linearity. Both properties play a central role in determining which solution methods apply.
The order of a differential equation is the highest derivative of the dependent variable that appears. Only derivatives, not exponents, affect the order.
A linear term includes exactly one occurrence of the dependent variable or one of its derivatives, raised to the first power and not inside another function. Coefficients do not affect linearity.
A differential equation is linear if all terms involving the dependent variable are linear. If even one is nonlinear, the entire equation is classified as nonlinear.
