# Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

## Section1.6Linear 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.

### Definition11.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.

### Note12.The linearity of a term depends entirely on the dependent variable.

The linearity of a term is determined by the presence of a single $$y$$ or derivative of $$y$$ to the power of $$1\text{.}$$ The coefficients, $$a(t)$$ play no part in the linearity.

### Example13.

$$\ \$$Given the differential equation,
\begin{equation*} e^{t}y^{(7)} + (t+1)y'y''' - t \ln y'' - y' \sin t - \tan y + \frac{4}{y} = \frac{3}{t}\text{,} \end{equation*}
identify each term and label the term as linear or nonlinear.
Solution.
To determine the linearity of a term we only need to consider the dependent variables (column $$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$$ Linearity dependson this column only
Note, since constant terms do not contain a dependent variable, they are necessarily linear. So, in the context of differential equations, $$\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.

### Example14.

$$\ \$$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*}

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.

$$y \sin(t)$$ is a linear term
• True
• Correct! The term $$\sin(t)y$$ is linear.
• False
• Incorrect. The term $$\sin(t)y$$ is indeed linear.

#### 3.$$t\sin(y')$$ is a linear term.

$$t\sin(y')$$ is a linear term
• True
• Incorrect, review the rules for identifying nonlinear terms.
• False
• Correct! A term that includes the derivative of the dependent variable inside another function is nonlinear.

#### 4.$$e^{ty}$$ is a linear term.

$$e^{ty}$$ is a linear term
• False
• Correct! The term $$e^{ty}$$ is nonlinear because the dependent variable $$y$$ is inside an exponential function.
• True
• Incorrect. Review the rules for identifying nonlinear terms.

#### 5.Identify the linear term in the equation: $$\ds\frac{1}{t}y'' + y^2 + \ln(y') = e^t$$.

Identify the linear term in the equation: $$\ds\frac{1}{t}y'' + y^2 + \ln(y') = e^t$$
• $$\frac{1}{t}y''$$
• Correct! The term $$\frac{1}{t}y''$$ is linear because it involves a single derivative of the dependent variable.
• $$y^2$$
• Incorrect. This term is nonlinear.
• $$\ln(y')$$
• Incorrect. This term is nonlinear because it involves the dependent variable inside a logarithmic function.
• $$e^t$$
• Incorrect. This term is linear because it involves the independent variable only.

#### 6.Which of the following is a nonlinear term?

Which of the following is a nonlinear term?
• $$\ds 3y$$
• Incorrect. This is a linear term.
• $$\ds t\sin(y')$$
• I The term $$t\sin(y')$$ is nonlinear because it involves a derivative inside a trigonometric function.
• $$\ds \frac{d^2y}{dx^2}$$
• Incorrect. This is a linear term.
• $$\ds 5y'$$
• Incorrect. This is a linear term.

#### 7.A linear term can contain the dependent variable multiplied by the independent variable.

A linear term can contain the dependent variable multiplied by the independent variable
• True
• Correct! For example, $$t y$$ is a linear term.
• False
• Incorrect. Carefully review the examples above.

#### 8.Which of the following terms is linear?

Which of the following terms is linear?
• $$y^2$$
• Incorrect. $$y^2$$ is nonlinear because the dependent variable is raised to a power other than one.
• $$\sin(t)y'$$
• Correct! $$\sin(t)y'$$ is linear because it is a function of the independent variable multiplied by the first derivative of the dependent variable.
• $$\ln(y)$$
• Incorrect. $$\ln(y)$$ is nonlinear because it includes the dependent variable inside another function.
• $$y y'$$
• Incorrect. $$y \cdot y'$$ is nonlinear because it involves the product of the dependent variable and its derivative.

#### 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.

Hint.

Hint.