DE-BOOK Differential Equation Components

Section Differential Equation Components

📝: 🎧 Listen.

Now that you can recognize what a differential equation looks like, it’s time to examine what they are made of. In this section, we’ll break down the core components and explain the role each part plays.

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Subsection Dependent and Independent Variables

Every differential equation involves two kinds of variables: a dependent variable and an independent variable. The dependent variable is the unknown function we’re solving for. It depends on the independent variable and always has derivatives applied to it with respect to that independent variable.

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For example, consider the equation:

\begin{equation} \frac{dy}{dx} + 2y = 4x^2.\tag{1} \end{equation}

Here, we’re solving for \(y\text{,}\) which makes it the dependent variable. Since we’re taking the derivative of \(y\) with respect to \(x\text{,}\) \(x\) is the independent variable.

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Identifying the dependent variable not only tells us what we’re solving for, but it will also help determine which methods we can use to solve the equation.

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🌌 Example 4. Identify the Variables.

Identify the dependent and independent variable in each equation:

\begin{equation*} \frac{dP}{ds} + \frac{P}{s^2} = 17s,\qquad u'' + t^2 u = 0,\qquad Q'' = 11Q \end{equation*}

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Solution.

The first equation contains \(P\) and \(s\text{,}\) but the presence of \(dP/ds\) implies \(P\) changes as \(s\) changes. So,

\(P\) is the unknown function we solve for that depends on \(s\text{.}\)

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dependent variable \(\rightarrow P\) , independent variable \(\rightarrow s\) .

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The middle equation contains \(u\) and \(t\text{,}\) but \(u''\) is the second derivative of \(u\text{.}\) So, \(u\) must change as \(t\) changes and

\(u\) is the unknown function we solve for that depends on \(t\text{.}\)

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dependent variable \(\rightarrow u\) , independent variable \(\rightarrow t\) .

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Only \(Q\) appears in the last equation, but \(Q''\) indicates that \(Q\) is changing and so it must be the dependent variable. Typically, the independent variable will be clear from context of the problem, but in this case just assume whatever variable you like.

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Checkpoint 5. 📖❓ Dependent and Independent Variables.

(a) 📖❓ Which Variable has the Derivative.

In a differential equation, you would expect to see a derivative applied to the dependent variable.

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True
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Correct! The dependent variable in a differential equation always has a derivative applied to it.
False
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Incorrect. By definition, a differential equation involves derivatives of the dependent variable.

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(b) 📖❓ Dependent vs. Independent.

Which variable in the differential equation,

\begin{equation*} \frac{dP}{ds} + \frac{P}{s^2} = 17s\text{,} \end{equation*}

represents the unknown function we would like to find?

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dependent variable, \(s\)
Incorrect. \(s\) is neither the dependent variable, nor what we are solving for.
independent variable, \(s\)
Incorrect! \(s\) is the independent variable, but it is not what we are solving for.
dependent variable, \(P\)
Yes! We are solving for the unknown, \(P\) which is the dependent variable in this equation.
independent variable, \(P\)
Incorrect. We are solving for \(P\text{,}\) but it is not the independent variable.

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(c) 📖❓ Identify the Unknown.

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Subsection Terms of a Differential Equation

📝: 🎧 Listen.

In math, the word “term” usually refers to a part of an expression that’s separated by plus or minus signs. The same idea applies to differential equations.

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So, a term in a differential equation is just any piece of the equation separated by addition or subtraction. For example, we might have something like:

\begin{equation*} \text{term}\ 1 + \text{term}\ 2 - \text{term}\ 3 = \text{term}\ 4 - \text{term}\ 5. \end{equation*}

This example has three terms on the left-hand side and two on the right.

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Terms in a differential equation can include the dependent variable, its derivatives, constants, or other functions of the independent variable. If a term doesn’t involve the dependent variable or its derivatives, we’ll call it a free term.

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📝: “Free” term.

For example, consider the five-term differential equation with dependent variable, \(y\text{:}\)

\begin{equation*} \frac{3}{x} y^{(6)} + 5.3 y'' + x^2 y' + y = \frac12\ln(x)\text{.} \end{equation*}

It is common to break the terms down based on the form of the dependent variable they contain. So, in terms of \(y\text{,}\) we have the following terms:

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\begin{equation} y^{(6)}\ \text{term} \end{equation}	\begin{equation} y''\ \text{term} \end{equation}	\begin{equation} y'\ \text{term} \end{equation}	\begin{equation} y\ \text{term} \end{equation}	\begin{equation} \text{free term} \end{equation}

\begin{equation} \frac{3}{x} y^{(6)} \end{equation}	\begin{equation} 5.3 y'' \end{equation}	\begin{equation} x^2 y' \end{equation}	\begin{equation} y \end{equation}	\begin{equation} \frac12\ln(x) \end{equation}

Every differential equation has a free term, even if it’s not written explicitly. For example, \(y' = -3y\) can be rewritten as \(y' + 3y = 0\text{,}\) where the free term is \(0\text{.}\)

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In general, any missing term can be interpreted as a zero term.

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Checkpoint 6. 📖❓ Terms of a Differential Equation.

(a) 📖❓ What is the Free Term?

Identify the free term in the differential equation

\begin{equation*} 3t^2 y' - 4t^2 = \frac{1}{t} y. \end{equation*}

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\(\quad 3t^2 y'\)
Incorrect. This term contains a derivative of the dependent variable \(y\text{,}\) so it is not a free term.
\(\quad \ds\frac{1}{t} y\)
Incorrect. This term involves the dependent variable \(y\text{,}\) so it is not a free term.
\(\quad 4t^2\)
Correct! \(4t^2\) is the free term because it does not contain the dependent variable \(y\text{.}\)

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(b) 📖❓ Hidden Free Term.

Identify the free term of the differential equation

\begin{equation*} w''=3tw. \end{equation*}

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\(\quad 3tw\)
Incorrect. This term involves the dependent variable \(w\text{,}\) so it is not a free term.
\(\quad 3t\)
Incorrect. This term involves the dependent variable \(w\text{,}\) so it is not a free term.
\(\quad 0\)
Correct! The free term is \(0\) because we can rewrite the equation as \(w'' - 3tw = 0\text{.}\)
This equation does not have a free term.
Incorrect. Every DE has a least one free term.

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Subsection Coefficients of a Differential Equation

📝: 🎧 Listen.

A coefficient in a differential equation is a constant or a function that depends only on the independent variable, and it multiplies the dependent variable or one of its derivatives. Coefficients help determine the structure of the equation and influence which solution methods are possible.

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For example, take the equation:

\begin{equation} y'' - 3y' + 2y = 0,\tag{2} \end{equation}

the coefficients of \(y''\text{,}\) \(y'\text{,}\) and \(y\) are the constants \(1\text{,}\) \(-3\text{,}\) and \(2\text{,}\) respectively.

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Now, coefficients don’t always have to be constants. They can also be functions of the independent variable. For instance, in the equation:

\begin{equation*} t^2 y'' + 5t y' + 6y = \sin(t), \end{equation*}

the coefficients are the following functions of \(t\text{:}\)

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\(\ul{y''\text{ coefficient}}\)	\(\ul{y'\text{ coefficient}}\)	\(\ul{y\text{ coefficient}}\)
\(t^2\)	\(5t\)	\(6\)

In general, for something to be considered a coefficient, it must not involve the dependent variable.

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Checkpoint 7. 📖❓ Coefficient of \(y'\).

Identify the coefficient of \(y'\) in the differential equation

\begin{equation*} 5y'' + 2\cos(t)y' - y = 7 \end{equation*}

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\(\quad \cos(t)\)
Incorrect, \(\cos(t)\) is only part of the coefficient of \(y'\text{.}\)
\(\quad 2\cos(t)\)
Correct! \(2\cos(t)\) is the coefficient of the term involving \(y'\text{.}\)
\(\quad 2\)
Incorrect, \(2\) is only part of the coefficient of \(y'\text{.}\)
\(\quad 7\)
Incorrect. \(7\) is the constant on the right-hand side of the equation.

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Subsection 📤 Wrap-Up

🗝️ \(\textbf{Key Takeaways...}\)

Every differential equation involves two variables: a dependent variable (the unknown function we’re solving for) and an independent variable (the variable it depends on).

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A term is any part of a differential equation separated by plus or minus signs. Terms can include derivatives, the dependent variable, constants, or other functions of the independent variable.

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A free term is any term that doesn’t involve the dependent variable or its derivatives, even a hidden \(0\) on one side of the equation counts.

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A coefficient is a constant or function of the independent variable that multiplies the dependent variable or its derivatives.

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Check Your Understanding.

Checkpoint 8. 🤔💭 Identify the Coefficients.

Click on each of the coefficients in the differential equation below.

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\(\phantom{vertical space hack - Is there a better way?}\)

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

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\(\phantom{vertical space hack - Is there a better way?}\)

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Hint.

Look for the dependent variable in each term. The coefficient is the constant or function that multiplies the dependent variable.

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You have attempted of activities on this page.

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\begin{equation} y^{(6)}\ \text{term} \end{equation}	\begin{equation} y''\ \text{term} \end{equation}	\begin{equation} y'\ \text{term} \end{equation}	\begin{equation} y\ \text{term} \end{equation}	\begin{equation} \text{free term} \end{equation}

\begin{equation} \frac{3}{x} y^{(6)} \end{equation}	\begin{equation} 5.3 y'' \end{equation}	\begin{equation} x^2 y' \end{equation}	\begin{equation} y \end{equation}	\begin{equation} \frac12\ln(x) \end{equation}