Section 1.6 Linear Terms
The most informative label you can place on a differential equation is whether it is linear or nonlinear. While linearity often refers to equations that graph as straight lines, the concept of linearity in differential equations is more nuanced. Understanding this distinction is crucial as it significantly influences how the equations are solved and interpreted.
Before jumping in, it may be helpful to revisit meaning of
dependent variables,
terms and
coefficients as we will be referencing them throughout this discussion.
Now, let’s define what it means for a term in a differential equation to be linear.
Definition 11. Linear Term.
- Linear Term
-
A term that has one of the forms
\begin{equation*}
a(t)\ y,\ a(t)\ y',\ a(t)\ y'',\ a(t)\ y''',\ \ldots
\end{equation*}
where \(y\) is the dependent variable. The coefficient, \(a(t)\text{,}\) can be a function of the independent variable, \(t\text{,}\) but not the dependent variable.
Note 12. The linearity of a term depends entirely on the dependent variable.
Example 13.
\(\ \ \)Given the differential equation,
\begin{equation*}
e^{t}y^{(7)} + (t+1)y'y''' - t \ln y'' - y' \sin t - \tan y + \frac{4}{y} = \frac{3}{t}\text{,}
\end{equation*}
identify each term and label the term as linear or nonlinear. Solution.
To determine the linearity of a term we only need to consider the dependent variables (column \(3\) below). We are looking for a single \(y\) or derivative of \(y\) raised to the power of \(1\text{.}\) If we find such a term, we label it as linear. Otherwise, it is nonlinear. Let’s separate the different parts in the following table.
| Term |
Coefficient |
Dependent Variables |
Linearity |
| \(\ds e^{t}\ y^{(7)}\) |
\(\ds e^{t}\) |
\(\ds y^{(7)}\) |
linear |
| \(\ds (t+1)y'y'''\) |
\(\ds (t+1)\) |
\(\ds y'y'''\) |
nonlinear |
| \(\ds t \ln y''\) |
\(\ds t\) |
\(\ds \ln y\) |
nonlinear |
| \(\ds y' \sin t\) |
\(\ds \sin t\) |
\(\ds y'\) |
linear |
| \(\ds \tan y\) |
\(\ds 1\) |
\(\ds \tan y\) |
nonlinear |
| \(\ds \frac{4}{y}\) |
\(\ds 4\) |
\(\ds \frac{1}{y}\) |
nonlinear |
|
|
\(\ds \uparrow\) |
|
|
|
Linearity depends
on this column only |
|
Note, since constant terms do not contain a dependent variable, they are necessarily linear. So, in the context of differential equations, \(\frac{3}{t}\) is also a linear term.
To help identify nonlinear terms, here are some common characteristics:
Identifying Nonlinear Terms.
A term is nonlinear if it contains a dependent variable, \(y\) or \(y^{(n)}\text{,}\) that is
Raised to a power other than 1, e.g., \(y^2\) or \((y')^{-4}\text{.}\)
Inside another function, e.g., \(\ln(y)\) or \(\sin(y)\text{.}\)
Multiplied or divided by another \(y\) or \(y^{(n)}\text{,}\) e.g., \(y y'''\) or \(\frac{y'}{y}\text{.}\)
Let’s practice this with one more example.
Example 14.
\(\ \ \)Determine the linearity of each term in the following differential equations.
\begin{equation*}
\frac{1}{t}y'' + y^2 + \ln(y') = e^t
\end{equation*}
Solution.
\begin{align*}
\underset{\text{linear}}{\underline{\frac{1}{t}{\color{blue} y'' }}} +
\underset{\text{nonlinear}}{\underline{{\color{blue} y}^2}} +
\underset{\text{nonlinear}}{\underline{\ln({\color{blue} y'})}} =\ \amp
\underset{\text{linear}}{\underline{e^t}}
\end{align*}
\begin{equation*}
P^{(6)} + \frac{m P'}{P} = (m - 1)^2
\end{equation*}
Solution.
\begin{align*}
\underset{\text{linear}}{\underline{P^{(6)}}} +
\underset{\text{nonlinear}}{\underline{\frac{m {\color{blue} P'}}{{\color{blue} P}}}} =\ \amp
\underset{\text{linear}}{\underline{(m - 1)^2}}
\end{align*}
Reading Questions Check your Understanding
Assume that \(y\) and \(t\) are the dependent and independent variables, respectively.
1. \(3t\) is a linear term.
\(3t\) is a linear term- True
- Correct! The term \(3t\) is linear because it is a function of the independent variable only.
- False
- Incorrect, review the examples of linear terms in the section on Linear Terms.
2. \(y \sin(t)\) is a linear term.
3. \(t\sin(y')\) is a linear term.
4. \(e^{ty}\) is a linear term.
5. Identify the linear term in the equation: \(\ds\frac{1}{t}y'' + y^2 + \ln(y') = e^t\).
6. Which of the following is a nonlinear term?
7. A linear term can contain the dependent variable multiplied by the independent variable.
8. Which of the following terms is linear?
9. Which of the following terms is linear?
Which of the following terms is linear?- \(\ds \frac{1}{t}y''\)
- Correct! \(\frac{1}{t}y''\) is linear because it is a function of the independent variable multiplied by the second derivative of the dependent variable.
- \(\ds y^3\)
- Incorrect. \(y^3\) is nonlinear because the dependent variable is raised to a power other than one.
- \(\ds e^t y^2\)
- Incorrect. \(e^t y^2\) is nonlinear because the dependent variable is squared.
- \(\ds y \cos(y)\)
- Incorrect. \(y \cdot \cos(y)\) is nonlinear because it involves the product of the dependent variable and a function of the dependent variable.
10. Which of the following is a characteristic of a nonlinear term?
Which of the following is a characteristic of a nonlinear term?- It involves the dependent variable raised to the first power.
- Incorrect. This is a characteristic of a linear term.
- It involves the dependent variable only as a constant.
- Incorrect. This is a characteristic of a linear term.
- It includes the dependent variable inside another function.
- Correct! A nonlinear term includes the dependent variable inside another function.
- It involves the independent variable only.
- Incorrect. This is a characteristic of a linear term.
11. Which term is an example of a nonlinear term?
Which term is an example of a nonlinear term?- \(3\)
- Incorrect. \(3\) is linear because it is a constant.
- \(3t\)
- Incorrect. \(3t\) is linear because it is a function of the independent variable only.
- \(y^2\)
- Correct! \(y^2\) is nonlinear because the dependent variable is squared.
- \(2t^2 y\)
- Incorrect. \(2t^2 y\) is linear because it is a function of the independent variable multiplied by the dependent variable.
12. Select the Linear Terms.
Click on all of the linear terms in the following differential equation.In this equation, \(\frac{d^2y}{dt^2}\text{,}\) \(t^2 y\text{,}\) \(\sin(t) y'\text{,}\) and \(3t\) are linear terms.
\(\displaystyle \frac{d^2y}{dt^2} \) \(\ +\ \) \(\displaystyle t^2 y \) \(\ +\ \) \(\displaystyle y^2 \) \(\ -\ \) \(\displaystyle \sin(t) y' \) \(\ =\ \) \(\displaystyle 3t \)
13. Select the Nonlinear Terms.
Click on all of the nonlinear terms in the following differential equation.In this equation, \(y^3\) and \(\ln(y)\) are nonlinear terms.
\(\displaystyle y^3 \) \(\ +\ \) \(\displaystyle e^t \frac{d^3y}{dt^3} \) \(\ -\ \) \(\displaystyle \ln(y) \) \(\ +\ \) \(\displaystyle t \frac{dy}{dt} \) \(\ +\ \) \(\displaystyle \frac{d^2y}{dt^2} \) \(\ =\ \) \(\displaystyle 0 \)
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