AP Calculus AB and BC glossary

Order of a differential equation

The order of a differential equation is the order of the highest derivative that appears in it. An equation containing only a first derivative is first order, and one containing a second derivative is second order. Every differential equation in AP Calculus is first order.

\frac{dy}{dx} = ky \quad \text{first order} \qquad \frac{d^{2}y}{dx^{2}} + 4y = 0 \quad \text{second order}

The order also counts the arbitrary constants in the general solution, so a first order equation carries one $C$ and needs exactly one initial condition to select a particular solution. A second order equation would need two, and those never appear on the AP exam.

Slope fields and Euler's method need the equation solved for $\frac{dy}{dx}$ as an expression in $x$ and $y$ , and separation of variables needs that expression to factor as a function of $x$ times a function of $y$ . First order is the weaker statement that no derivative past the first appears, which is why the squared example below still counts as first order. The CED has you solve algebraically only by separation of variables, in topics 7.6 and 7.7, and hands everything else to slope fields, plus Euler's method in BC.

The mistake

Confusing order with degree. In $\left(\frac{dy}{dx}\right)^{2} = x$ the highest derivative is still the first one, so the equation is first order even though the derivative is squared. Degree counts the power, order counts the differentiations.

Appears in: Unit 7: Differential Equations