Section 1.7 Linearity
In the world of differential equations, distinguishing between linear and nonlinear equations is vital. This distinction will often guide you on how to approach solving the equation and what methods to use. The good news is, all the work we did in the previous section will make this task much easier.
Definition 15.
A differential equation is linear if it contains only linear terms. That is, it can be expressed as
\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}} =
\underset{\text{linear term}}{\underbrace{f(t).}}\tag{3}
\end{equation}
If just one of the terms in the equation is nonlinear, then the entire differential equation is nonlinear.This definition might seem abstract at first, but it encompasses all possible linear differential equations. In practice, most equations you’ll encounter will involve only a few terms, and the challenge lies in identifying whether any of those terms break the rule of linearity. If they do, the equation is nonlinear.
Example 16.
\(\ \ \)Let’s determine whether the following differential equation is linear:
\begin{equation*}
y'' + \frac{y'}{t^2} + y = 17t
\end{equation*}
Solution.
To classify this equation, we need to verify if every term involving \(y\) and its derivatives is linear. Let’s break it down term by term:
\begin{gather*}
y'' + \frac{y'}{t^2} + y =\ 17t \\
\underset{\text{linear}}{\underline{\color{blue}{y''}}} +
\underset{\text{linear}}{\underline{\color{blue}{\frac{1}{t^2} y'}}} +
\underset{\text{linear}}{\underline{\color{blue}{y}}} =
\underset{\text{constant term}}{\underline{\color{blue}{17t}}}
\end{gather*}
Since each term involving \(y\) or its derivatives is linear, this differential equation is indeed linear.
In summary, recognizing whether a differential equation is linear or nonlinear will help you determine the appropriate methods for solving it. Linear equations allow for a more straightforward approach, while nonlinear equations often require specialized techniques.
Reading Questions Check your Understanding
1. The differential equation \(\frac{dy}{dt} + t^2 y = e^t\) is linear.
2. The differential equation, \(\ds y'' + y' \cos t = 7y \text{,}\) is .
The differential equation, \(\ds y'' + y' \cos t = 7y \text{,}\) is
 Linear
Yes! This DE is linear. Each term involving \(y\) or its derivatives, such as \(y''\text{,}\) \(y' \cos t\text{,}\) and \(7y\text{,}\) is linear. A linear differential equation contains terms where the dependent variable and its derivatives appear to the first power and are not multiplied by each other.
 Nonlinear

No, this is linear. Looking carefully at each term, we see:
\begin{gather*}
y'' + y' \cos t = 7y \\
\underset{\text{linear}}{\underline{(1){\color{blue} y'' }}} +
\underset{\text{linear}}{\underline{(\cos t){\color{blue} y' }}} =
\underset{\text{linear}}{\underline{7{\color{blue} y}}}
\end{gather*}
Since every term is linear, this differential equation is linear. Review the definition of linear differential equations.
3. The differential equation, \(\ds y'' + \sin(y) = 17t \text{,}\) is .
The differential equation, \(\ds y'' + \sin(y) = 17t \text{,}\) is
 Linear

No, this is nonlinear. Looking carefully at each term, we see:
\begin{gather*}
y'' + \sin(y) = 17t \\
\underset{\text{linear}}{\underline{(1){\color{blue} y'' }}} +
\underset{\text{nonlinear}}{\underline{\sin({\color{blue} y})}} =
\underset{\text{linear}}{\underline{17{\color{blue} t}}}
\end{gather*}
Since one term is not linear, this differential equation is nonlinear. Revisit the rules for linearity and nonlinearity.
 Nonlinear
Yes! This DE is nonlinear since the \(\sin(y)\) term is not linear. Nonlinear terms involve functions like sine, logarithms, or powers greater than one when applied to the dependent variable \(y\text{.}\)
4. The differential equation, \(\ds \frac{d^2x}{dt^2} + e^x = 0 \text{,}\) is .
The differential equation, \(\ds \frac{d^2x}{dt^2} + e^x = 0 \text{,}\) is
 Linear
 Incorrect. The term \(e^x\) makes this equation nonlinear, as it involves the exponential function of the dependent variable.
 Nonlinear
 Correct! The term \(e^x\) introduces nonlinearity into the equation, as it involves the dependent variable \(x\) inside an exponential function.
5. Select the linear differential equation.
6. Which term makes the equation \(\ds y''' + 3y' \sin(t) + y^2 = 0\) nonlinear?
7. Select all the linear differential equations.
Select all the linear differential equations
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.
\(\ds y'' + \sin(y) = 17t \) 
\(\ds y'' + \frac{y'}{t^2} + y = 17t \) 
\(\ds y'' + 3y' + 2y = 0 \) 
\(\ds y'' + y^2 = 17t \) 
\(\ds y'' + \frac{y'}{t} + y = 17t \) 
\(\ds y = y' \) 
Hint.
Remember that a linear differential equation contains only linear terms. Four of these equations are linear.
8. Select all the nonlinear differential equations.
Click on all of the linear differential equations.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.
\(\ds \frac{dx}{ds} = x^2  4 \) 
\(\ds \frac{d^2u}{dz^2}  5 \frac{du}{dz} + 6u = 0 \) 
\(\ds \frac{dp}{d\tau} + \sin(p) = \tau^2 \) 
\(\ds \frac{dw}{dv} + 2vw = \cos(v) \) 
\(\ds \frac{dr}{d\theta} + r^3 = \theta \) 
\(\ds \frac{dN}{dt} = kN \) 
\(\ds \frac{dm}{dq} = m^3  q^2 \) 
\(\ds \frac{dz}{dt} + z\frac{dz}{dt} = t^3 \) 
\(\ds \frac{dy}{dx} = y \ln(y) \) 
Hint.
First identify the dependent variable, then carefully look at each term to determine if it is nonlinear.
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