Section 1.4 Terms & Coefficients
This text will frequently refer to “terms” and “coefficients”. Here is the definition.
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.
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.
Reading Questions Check your Understanding
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.
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.
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.
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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