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

## Section1.4Terms & Coefficients

This text will frequently refer to “terms” and “coefficients”. Here is the definition.

### Definition6.Terms & Coefficients.

In differential equations, terms and coefficients are defined as follows:
Terms
The expressions separated by $$+\text{,}$$ $$-\text{,}$$ or $$=$$ signs.
Coefficients
The objects multiplied by the dependent variable or one of its derivatives.
Constant Term
A term without a dependent variable is called a constant term and is not a coefficient.
Consider the differential equation:
$$\us{y^{(6)} \text{ term} }{\ub{\ \frac{3}{x} {\color{blue}\ y^{(6)} } } } + \us{y'' \text{ term} }{\ub{\ 5.3 {\color{blue}\ y'' } } } + \us{y' \text{ term} }{\ub{\ x^2 {\color{blue}\ y' } } } - \us{y \text{ term} }{\us{\uparrow}{ {\color{blue}\ \ul{y} } } } = \us{\text{constant term} }{\ub{\ \frac12\ln(x)\ } }\text{.}\tag{2}$$
This equation has five terms and four coefficients: $$\frac{3}{x}\text{,}$$ $$5.3\text{,}$$ $$x^2\text{,}$$ and $$-1\text{.}$$ Notice that coefficients can be functions of the independent variable (like $$\frac{3}{x}$$ and $$x^2$$) or constants (like $$5.3$$ and $$-1$$). The distinction between constant and variable coefficients will become crucial when we study a group of differential equations known as constant-coefficient equations.

### Example7.Terms & Coefficients in a Differential Equation.

Identify the terms and coefficients of the differential equation
\begin{equation*} 3t^2\ y' - 4\cos t + \frac{y'y}{t} - 515 y = 0 \end{equation*}
Solution.
The equation can be broken down as follows:
\begin{equation*} \us{y' \text{ term}}{\ub{\ 3t^2{\color{blue} y'}\ }} - \us{\text{constant term}}{\ub{\ 4\cos t\ }} + \us{y'y \text{ term}}{\ub{\ \frac{1}{t}{\color{blue} y'y}\ }} - \us{y \text{ term}}{\ub{\ 515{\color{blue} y}\ }} = 0\text{.} \end{equation*}
The coefficients are $$3t^2\text{,}$$ $$\frac{1}{t}\text{,}$$ and $$-515\text{.}$$ Notice that $$3t^2$$ and $$\frac{1}{t}$$ are functions of the independent variable $$t\text{,}$$ whereas $$-515$$ is a constant.

For the following, assume $$y$$ is the dependent variable as a function of $$t\text{.}$$

#### 1.Given $$\ds 5y'' + 2y' - \cos(t) y = 7\text{,}$$ what is the coefficient of $$\ds y'\text{?}$$

Given $$\ds 5y'' + 2y' - \cos(t) y = 7\text{,}$$ what is the coefficient of $$\ds y'\text{?}$$
• $$5$$
• Incorrect. $$5$$ is the coefficient of $$y''\text{.}$$
• $$2$$
• Correct! $$2$$ is the coefficient of the term involving $$y'\text{.}$$
• $$\cos(t)$$
• Incorrect. $$\cos(t)$$ is the coefficient of $$y\text{.}$$
• $$7$$
• Incorrect. $$7$$ is the constant on the right-hand side of the equation.

#### 2.Given $$\ds 3t^2 y' + \frac{1}{t} y - 4 = 0\text{,}$$ which of the following is considered a constant term?

Given $$\ds 3t^2 y' + \frac{1}{t} y - 4 = 0\text{,}$$ which of the following is considered a constant term?
• $$3t^2 y'$$
• Incorrect. This term contains a derivative of the dependent variable $$y\text{,}$$ so it is not a constant term.
• $$\frac{1}{t} y$$
• Incorrect. This term involves the dependent variable $$y\text{,}$$ so it is not a constant term.
• $$-4$$
• Correct! $$-4$$ is the constant term because it does not depend on the dependent variable $$y$$ or its derivatives.

#### 3.$$3t$$ is an example of a constant term.

$$3t$$ is an example of a constant term
• True
• Correct! In the context of differential equations, $$3t$$ is a constant term since it is not multiplied by the dependent variable $$y$$ or one of its derivatives.
• False
• Incorrect. While $$3t$$ is not a constant function, it is a constant term in the context of differential equations.

#### 4.$$y$$ is the coefficient of the term $$y \sin(t)$$.

$$y$$ is the coefficient of the term $$y \sin(t)$$
• True
• Incorrect. The coefficient is the factor multiplying the entire term involving the dependent variable, not the dependent variable itself.
• False
• Correct! The coefficient is what multiplies the term involving the dependent variable, so in this case, the coefficient of $$y \sin(t)$$ is $$\sin(t)\text{,}$$ not $$y\text{.}$$

#### 5.The term $$\ds y'''$$ does not have a coefficient.

The term $$\ds y'''$$ does not have a coefficient
• True
• Incorrect. Every term in a differential equation has a coefficient, even if that coefficient is simply 1.
• False
• Correct! The coefficient of $$y'''$$ is 1, even if it is not explicitly written.

#### 6.Given $$\ds e^t y''' + 4y' - 3y = \sin(t)\text{,}$$ which terms has a function as its coefficient?

Given $$\ds e^t y''' + 4y' - 3y = \sin(t)\text{,}$$ which terms has a function as its coefficient?
• $$e^t y'''$$
• Correct! $$e^t$$ is a function of $$t$$ and acts as the coefficient of $$y'''\text{.}$$
• $$4y'$$
• Incorrect. $$4$$ is a constant coefficient, not a function.
• $$-3y$$
• Incorrect. $$-3$$ is a constant coefficient, not a function.
• $$\sin(t)$$
• Incorrect. $$\sin(t)$$ is on the right-hand side of the equation and is not acting as a coefficient for any term.

#### 7.Given $$\ds t^3 y'' + 6 y' - \ln(t) y = 0\text{,}$$ which statement best describes the coefficient of $$y\text{?}$$

Given $$\ds t^3 y'' + 6 y' - \ln(t) y = 0\text{,}$$ which statement best describes the coefficient of $$y\text{?}$$
• It is a constant coefficient
• Incorrect. A constant coefficient does not depend on the independent variable.
• It is a function of the independent variable
• Correct! The coefficient $$\ln(t)$$ depends on the independent variable $$t\text{.}$$
• There is no coefficient
• Incorrect. The term $$\ln(t) y$$ has a coefficient, which is $$\ln(t)\text{.}$$
• It is an arbitrary constant
• Incorrect. $$\ln(t)$$ is a specific function of $$t\text{,}$$ not an arbitrary constant.

#### 8.Given $$\ds\frac{d^2y}{dt^2} - 3t^2 y' + 4y = 0\text{,}$$ which of the following statements is true?

Given $$\ds\frac{d^2y}{dt^2} - 3t^2 y' + 4y = 0\text{,}$$ which of the following statements is true?
• The coefficient of $$y'$$ is $$-3t^2\text{.}$$
• Correct! The term $$-3t^2 y'$$ has a coefficient of $$-3t^2\text{.}$$
• The coefficient of $$y$$ is $$-4\text{.}$$
• Incorrect. The coefficient of $$y$$ is $$4\text{,}$$ not $$-4\text{.}$$
• The coefficient of $$y'$$ is $$-3t\text{.}$$
• Incorrect. The correct coefficient of $$y'$$ is $$-3t^2\text{,}$$ not $$-3t\text{.}$$
• There is no constant term in the equation.
• Incorrect. The equation does not include a constant term since all terms involve the dependent variable or its derivatives.

#### 9.Select all the coefficients in the differential equation.

Hint.
Review the example in this section for more guidance on identifying coefficients.