DE-BOOK Order & Linearity

Section Order & Linearity

So far, we’ve learned how to identify the core parts of a differential equation, its variables, terms, and coefficients. Now we’ll turn to how differential equations are classified based on how these components appear. In this section, we’ll focus on two of the most important classifications: the equation’s order, which tells us the highest derivative involved, and its linearity, which depends on how the dependent variable and its derivatives appear in each term. Both properties play a central role in determining what kinds of solution methods apply.

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Subsection Order

A differential equation must contain at least one derivative, but there’s 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.

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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.

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Checkpoint 9. 📖❓ What is the Order.

The order of a differential equation is determined by the number of terms it contains

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True
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Incorrect. The order is based on the highest derivative, regardless of the number of terms.
False
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Correct! The order is determined by the highest derivative, not the number of terms.

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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 three derivatives in the \(y'''\) term.

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Checkpoint 10. 📖❓ Select the Third-Order Equation.

Which of the following equations is a third-order differential equation?

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\(\quad \ds\frac{d^3y}{dx^3} + x^2y = 0\)
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Correct! The highest derivative here is the third derivative, making it a third-order differential equation.
\(\quad \ds\frac{d^2y}{dx^2} + y' = \sin x\)
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Incorrect. This is a second-order differential equation.
\(\quad y'' + y' + y = 0\)
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Incorrect. This is a second-order differential equation.
\(\quad y' + y = x\)
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Incorrect. This is a first-order differential equation.

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🌌 Example 11. 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*}

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Solution.

The first equation is second-order because the highest derivative of \(A\) is \(\frac{d^2 A}{dt^2}\text{.}\)

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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.

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So, the second equation is fifth-order. The fact that it appears inside a sine function makes no difference to the order.

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Checkpoint 12. 📖❓ Give the Order of Each Differential Equation.

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Subsection Linear Terms

To decide whether a differential equation is linear, we first need to understand what it means for a term to be linear. In this context, linearity refers to how the dependent variable and its derivatives appear in each term.

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Let \(y\) be the dependent variable. A term is linear if it appears as either:

\begin{equation} a(t), \quad a(t)y, \quad \text{or} \quad a(t)y^{(n)},\tag{3} \end{equation}

where the coefficient, \(a(t)\text{,}\) can be a constant or a function of the independent variable \(t\) only, and \(y^{(n)}\) is the \(n\)-th derivative of \(y\text{.}\)

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For example, the following terms are linear:

\begin{equation*} \us{a(t)}{\us{\uparrow}{\ul{23}}}, \quad \us{a(t)}{\us{\uparrow}{\ul{2t^3}}}y'', \quad \us{a(t)}{\us{\uparrow}{\ul{\frac{1}{t}}}}y^{(5)}, \quad \us{a(t)}{\us{\uparrow}{\ul{e^{2t}}}}y, \quad \us{a(t)}{\us{\uparrow}{\ul{7\sin(t)}}}y'. \end{equation*}

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The coefficient of the term actually has no affect on whether it’s linear, regardless of how complicated \(a(t)\) appears.

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Checkpoint 13. 📖❓ Identify the Coefficient.

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Subsection Nonlinear Terms

A term is nonlinear if it not in one of the forms in (3). Assuming \(y\) is the dependent variable, some telltale signs that a term is nonlinear include:

\(y\) or \(y^{(n)}\) are raised to a power other than 1
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e.g., \(\ y^2,\ (y')^{-4}\)

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\(y\) or \(y^{(n)}\) appear inside a nonlinear function
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e.g., \(\ \ln(y),\ \sin(y),\ e^{y'}\)

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\(y\) or \(y^{(n)}\) are multiplied or divided by each other
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e.g., \(\ y y''',\ \sfrac{y'}{y}\)

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Let’s apply these strategies by breaking down the linearity of each term in the following differential equation:

\begin{equation*} y\frac{d^5y}{dt^5} + 2t^3\ \frac{d^2y}{dt^2} + \ln\left(\frac{dy}{dt}\right) + y^2 = t^3 \end{equation*}

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The terms are:

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\(\ds y\frac{d^5y}{dt^5}\) 🔗: Nonlinear. This term involves a product of the dependent variable and its derivative, violating the rule that only one can appear in a term.
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\(\ds 2t^3\frac{d^2y}{dt^2}\) 🔗: Linear. The derivative \(y''\) appears by itself and to the first power. The coefficient \(2t^3\) does not affect linearity.
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\(\ds \ln\left(\frac{dy}{dt}\right)\) 🔗: Nonlinear. The derivative \(y'\) appears inside a nonlinear function, the natural logarithm.
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\(\ds y^2\) 🔗: Nonlinear. The dependent variable appears raised to a power other than one.
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\(\ds t^3\) 🔗: Linear. This term doesn’t involve the dependent variable or any of its derivatives.
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Checkpoint 14. 📖❓ Match Each Terms with its Linearity.

Directions:
🔹 🔗	Match (by dragging) each term (on the left) to either Linear or Nonlinear.
🔹 🔗	Assume the dependent variable is \(y\) and the independent variable is \(t\text{.}\)

\(3y''\)
\(t^2y\)
\(y' \cos t\)
\(3/t\)
Linear
\(y^2\)
\(\sin(y')\)
Non-Linear Term

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See if you can identify the linearity of the terms in the following examples, before looking at the solutions.

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🌌 Example 15. Identify the Linearity of the Terms.

Determine whether each term in the equation is linear or nonlinear:

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\(\ds\frac{1}{t}y'' + y^2 + \ln(y') = e^t\)

Solution.

Focus on the dependent variable, \({\color{BurntOrange} y}\text{,}\) and its derivatives.

\begin{gather*} \underset{\overline{\text{linear}}}{\frac{1}{t}{\color{BurntOrange} y''}} + \underset{\overline{\text{nonlinear}}}{{\color{BurntOrange} y}^2} + \underset{\overline{\text{nonlinear}}}{\ln({\color{BurntOrange} y'})} =\ \underset{\overline{\text{linear}}}{e^t} \end{gather*}

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\(\ds P^{(6)} + \frac{m P'}{P} = (m - 1)^2\)

Solution.

Focus on the dependent variable, \({\color{BurntOrange} P}\text{,}\) and its derivatives.

\begin{gather*} \underset{\text{linear}}{\underline{\color{BurntOrange} P^{(6)}}} + \underset{\overline{\text{nonlinear}}}{\frac{m {\color{BurntOrange} P'}}{{\color{BurntOrange} P}}} =\ \underset{\text{linear}}{\underline{(m - 1)^2}} \end{gather*}

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Checkpoint 16. 📖❓ Identify the Nonlinear Terms.

For each differential equation, click on all of the nonlinear terms.

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\(\ds \frac{d^2y}{dt^2} \) \(\ +\ \) \(\ds t^2 y \) \(\ +\ \) \(\ds y^2 \) \(\ -\ \) \(\ds \sin(t) y' \) \(\ =\ \) \(\ds 3t \)

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\(\ds tyy' \) \(\ +\ \) \(\ds e^t \frac{d^3y}{dt^3} \) \(\ -\ \) \(\ds \ln(y) \) \(\ +\ \) \(\ds \frac{y}{t} \) \(\ +\ \) \(\ds \frac{t}{y}\frac{d^2y}{dt^2} \) \(\ =\ \) \(\ds 0 \)

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Subsection Linearity

Now that we’ve defined what it means for a term to be linear, we can describe what it means for an entire differential equation to be linear.

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Essentially, a differential equation is linear if every term in the equation is linear. If just one term is nonlinear, the entire equation is nonlinear. We formally define linear differential equations as follows:

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Checkpoint 17. 📖❓ What Makes an Equation Linear of Nonlinear.

Select the true statement.

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An equation is nonlinear if at least one of its terms are nonlinear.
Correct! A differential equation is nonlinear if any of its terms are nonlinear.
An equation is linear if at least one of its terms are linear.
Incorrect. A differential equation needs all of its terms to be linear.
As long as there are more linear terms than nonlinear terms, the equation is linear.
Incorrect.

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📙 Definition 18. Linear Differential Equations.

A differential equation is linear if it can be written as a sum of linear terms involving the dependent variable and its derivatives:

This structure is known as a linear combination of the dependent variable and its derivatives.

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\begin{equation} \underset{\text{linear term}}{\underbrace{a_n(t)\ y^{(n)}}} + \cdots + \underset{\text{linear term}}{\underbrace{a_2(t)\ y''}} + \underset{\text{linear term}}{\underbrace{a_1(t)\ y'}} + \underset{\text{linear term}}{\underbrace{a_0(t)\ y}} = f(t)\tag{4} \end{equation}

where each \(a_k(t)\) is a function of the independent variable only, and \(f(t)\) is the input or forcing term.

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For example, the following differential equations are linear:

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\begin{equation} 3\frac{dy}{dx} + 2y = 0, \qquad y''' - 2y' - \frac{7}{x}y = e^x, \qquad w'' - \tan(t) = 3t.\tag{5} \end{equation}

Each term involving the dependent variable or its derivatives is linear, none appear raised to a power, inside a function, or multiplied together.

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In contrast, the following equations are nonlinear because they include at least one nonlinear term (✷):

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\begin{equation} y' + \us{\large ✷}{\ul{ t y^2 }} = t, \qquad t \frac{dP}{dx} + \us{\large ✷}{\ul{ \sin(P) }} = t^2, \qquad \us{\large ✷}{\ul{\omega\omega^{(5)}}} + s = \us{\large ✷}{\ul{ e^\omega }}\tag{6} \end{equation}

In each case, it only took one term to violate the definition of a linear differential equation.

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Let’s practice identifying the linearty of entire equations with a few examples.

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🌌 Example 19. Identify the Linearity of the Equation.

Determine whether each of the following differential equations is linear:

\begin{equation*} y^{(5)} + \frac{y'}{x^2} + 5y = \ln x, \qquad P^{(6)} + \frac{m P'}{P} = (m - 1)^2 \end{equation*}

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Solution.

In the first equation, all terms involving \(y\) or its derivatives appear linearly:

\begin{gather*} \underset{\overline{\text{linear}}}{\vphantom{\frac11}\color{BurntOrange} y^{(5)}} + \underset{\overline{\text{linear}}}{\frac{1}{x^2} \color{BurntOrange} y'} + \underset{\overline{\text{linear}}}{5 \color{BurntOrange} y} =\ \underset{\overline{\text{linear}}}{\ln x} \end{gather*}

So the equation is linear.

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In the second equation, the term \(\frac{m P'}{P}\) is nonlinear, since \(P\) appears in both the numerator and denominator:

\begin{gather*} \underset{\text{linear}}{\underline{\color{BurntOrange} P^{(6)}}} + \underset{\overline{\text{nonlinear}}}{\frac{m \color{BurntOrange} P'}{\color{BurntOrange} P}} =\ \underset{\text{linear}}{(m - 1)^2} \end{gather*}

This makes the entire equation nonlinear.

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📤 Wrap-Up.

🗝️ Key Takeaways...

📝: 🎧 Listen.

The order of a differential equation is the highest derivative of the dependent variable that appears. Only derivatives, not exponents, affect the order.

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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.

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A nonlinear term involves a product, quotient, exponent, or function of the dependent variable or its derivatives.

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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.

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Check Your Understanding.

Checkpoint 20. 📖❓ Linear System Basics.

(a) Identify the Linear Equations.

Click-on all the linear differential equations.

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Linear equations only involve the dependent variable and its derivatives to the first power, and they won’t be inside nonlinear functions like sine or multiplied by each other.

\(\ds y'' + \sin(y) = 17t \)	\(\ds y'' + \frac{y'}{t^2} + y = 17t \)	\(\ds y'' + 3y' + 2y = 0 \)
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\(\ds y'' + y^2 = 17t \)	\(\ds y'' + \frac{y'}{t} + y = 17t \)	\(\ds y = y' \)

Hint.

Remember that a linear differential equation contains only linear terms. Four of these equations are linear.

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(b) Identify the Nonlinear Equations.

Click-on all the nonlinear differential equations

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Nonlinear equations often have terms where the dependent variable or its derivatives are raised to powers other than one, or are inside functions like sine, logarithms, or are multiplied by each other.

\(\ds \frac{dx}{ds} = x^2 - 4 \)	\(\ds \frac{d^2u}{dz^2} - 5 \frac{du}{dz} + 6u = 0 \)	\(\ds \frac{dp}{d\tau} + \sin(p) = \tau^2 \)
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\(\ds \frac{dw}{dv} + 2vw = \cos(v) \)	\(\ds \frac{dr}{d\theta} + r^3 = \theta \)	\(\ds \frac{dN}{dt} = -N \)
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\(\ds \frac{dm}{dq} = m^3 - q^2 \)	\(\ds \frac{dz}{dt} + z\frac{dz}{dt} = t^3 \)	\(\ds \frac{dy}{dx} = y \ln(y) \)

Hint.

First identify the dependent variable, then carefully look at each term to determine if it is nonlinear. There are six nonlinear equations here.

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You have attempted of activities on this page.

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