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Section 1.4 Terms & Coefficients

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

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Definition 6. 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:

\begin{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}
\end{equation}

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*.

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Example 7. *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.

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Reading Questions Check your Understanding

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

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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.

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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.

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3. *\(3t\) is an example of a constant term*.

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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 \(y\text{.}\)

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5. *The term \(\ds y'''\) does not have a coefficient*.

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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.

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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.

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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.

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

*Select all the coefficients in the differential equation*

The coefficients in this equation are \(t^2\text{,}\) \(4\text{,}\) and \(t\text{.}\) Remember, coefficients are the factors multiplied by the dependent variable or its derivatives.

\(t \) \(\displaystyle \frac{d^2y}{dt^2} \) \(\ +\ \) \(\displaystyle t^2 \) \(\displaystyle y^2 \) \(\ -\ \) \(\displaystyle 4 \) \(\displaystyle y' \) \(\ =\ \) \(\displaystyle \frac{1}{y} \) \(\displaystyle t \) \(\ +\ \) \(\displaystyle \sin(t) \)

## Hint.

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

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