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Section 2.1 Order
A
differential equation must contain at least one derivative, but there is no limit on how many derivatives can appear. Some equations involve just a first derivative, while others include higher-order derivatives like
\(y''\text{,}\) \(y^{(3)}\text{,}\) or beyond.
To capture this idea, we define the
order of a differential equation as the highest derivative of the dependent variable that appears in the equation.
Checkpoint 11 . 📖❓ Order and Number of Terms.
The order of a differential equation is determined by the number of terms it contains.
True
Incorrect. The order is based on the highest derivative, regardless of the number of terms.
False
Correct! The order is determined by the highest derivative, not the number of terms.
For example, the equation
\begin{equation*}
\frac{dy}{dx} + y = x
\end{equation*}
is a
first-order differential equation because the highest derivative is
\(\frac{dy}{dx}\text{.}\) On the other hand, the differential equation
\begin{equation*}
x^2y'' + y''' = \sin(x) + y^8
\end{equation*}
is third-order because it contains up to the third derivative in the \(y'''\) term.
Checkpoint 12 . 📖❓ Spot the Third-Order Equation.
Which of the following equations is a third-order differential equation?
\(\quad \ds\frac{d^3y}{dx^3} + x^2y = 0\)
Correct! The highest derivative here is the third derivative, making it a third-order differential equation.
\(\quad \ds\frac{d^2y}{dx^2} + y' = \sin x\)
Incorrect. This is a second-order differential equation.
\(\quad y'' + y' + y = 0\)
Incorrect. This is a second-order differential equation.
\(\quad y' + y = x\)
Incorrect. This is a first-order differential equation.
🌌 Example 13 . Identify the Order.
For each of the following differential equations, identify the order:
\begin{equation*}
\frac{d^2 A}{dt^2} + \frac{dA}{dt} + A = 17, \qquad w^6 + \sin\big(w^{(5)}\big) = 0
\end{equation*}
Solution .
The first equation is
second-order because the highest derivative of
\(A\) is
\(\frac{d^2 A}{dt^2}\text{.}\)
A common pitfall for beginners is to confuse exponents with derivatives. In the second equation, the exponent in the term
\(w^6\) refers to
\(w\) raised to the sixth power, not a sixth derivative. Only derivatives affect the order.
So, the second equation is
fifth-order . The fact that it appears inside a sine function does not affect the order.
Checkpoint 14 . 📖❓ Give the Order.
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activities on this page.
