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Exercises 2.5 Exercises
💡 Conceptual Quiz.
Answer the following questions to test your understanding of the concepts.
1. True or False.
Select True or False for each statement.
(a)
In a differential equation, the dependent variable always has at least one derivative applied to it.
True
Correct! A differential equation always contains at least one derivative of the dependent variable.
False
Incorrect. Even if \(y\) appears without a derivative, a differential equation still includes at least one derivative of \(y\text{.}\)
(b)
A linear term can contain the dependent variable multiplied by the independent variable.
2. Multiple Choice.
(a)
Which of the following equations is a fourth-order differential equation?
\(\quad y^{(4)} - y = e^x\)
Correct! The highest derivative here is the fourth derivative, making it a fourth-order differential equation.
\(\quad y''' + 2y = 0\)
Incorrect. This is a third-order differential equation.
\(\quad y'' + 3y' = x\)
Incorrect. This is a second-order differential equation.
\(\quad y' - 5y = e^x\)
Incorrect. This is a first-order differential equation.
(b)
Which term is an example of a nonlinear term?
\(\quad 3\)
Incorrect. \(3\) is linear because it is a constant.
\(\quad 3t\)
Incorrect. \(3t\) is linear because it is a function of the independent variable only.
\(\quad y^2\)
Correct! \(y^2\) is nonlinear because the dependent variable is squared.
\(\quad 2t^2 y\)
Incorrect. \(2t^2 y\) is linear because it is a function of the independent variable multiplied by the dependent variable.
(c)
Which term makes the equation
\(y''' + 3y' \sin(t) + y^2 = 0\) nonlinear?
\(y^2\)
Correct! The term \(y^2\) is nonlinear because the dependent variable \(y\) is raised to the second power.
\(3y' \sin(t)\)
Incorrect. While this term includes a function of \(t\text{,}\) it is still linear because \(y'\) appears to the first power.
\(y'''\)
Incorrect. The term \(y'''\) is linear because \(y\) and its derivatives are to the first power.
(d)
Which of the following describes an example of a
nonlinear term?
A dependent variable inside another function.
Correct! This would be an example of a nonlinear term.
A dependent variable raised to the first power.
Incorrect. This is a characteristic of a linear term.
A dependent variable multiplied by a constant.
Incorrect. This is a characteristic of a linear term.
An independent variable squared.
Incorrect. The linearity of a term only depends on the dependent variable.
3. Matching.
(a)
Consider the differential equation
\begin{equation*}
y'' + y' \cos t = 7e^y\text{.}
\end{equation*}
Drag each expression (left) to the appropriate label (right).
\(y\) Dependent Variable
\(t\) Independent Variable
\(y' \cos t\) Linear Term
\(7e^y\) Nonlinear Term
\(2\) Order of the DE
\(1\) Coefficient of \(y''\)
\(\cos t\) Coefficient of \(y'\)
Answer .
Dependent variable:
\(y\text{.}\) Independent variable:
\(t\text{.}\) Linear term:
\(y'\cos t\text{.}\) Nonlinear term:
\(7e^y\text{.}\) Order:
\(2\text{.}\) Coefficient of
\(y''\text{:}\) \(1\text{.}\) Coefficient of
\(y'\text{:}\) \(\cos t\text{.}\)
🏋️♂️ Practice Drills.
More practice to strengthen your understanding.
4. Identify the Linear & Nonlinear Terms.
(a)
Click on all of the linear terms in the differential equation.
The linear terms are those where \(y\) or its derivatives appear only to the first power.
\(\vphantom{y'}\)
\(\dfrac{d^2y}{dt^2} \) \(\ +\ \) \(t^2 y \) \(\ +\ \) \(y^2 \) \(\ -\ \) \(\sin(t) y' \) \(\ =\ \) \(3t \)
\(\vphantom{y'}\)
Answer .
Linear terms:
\(\dfrac{d^2y}{dt^2}\text{,}\) \(t^2y\text{,}\) \(\sin(t)y'\text{,}\) and
\(3t\text{.}\)
(b)
Identify the nonlinear terms in the differential equation:
\begin{equation*}
yy'' + y^2 + \ln(y') = e^t
\end{equation*}
\(\quad yy''\)
Nonlinear. This term contains the product of the dependent variable and its second derivative.
\(\quad y^2\)
Nonlinear. This term contains the dependent variable squared.
\(\quad \ln(y')\)
Nonlinear. The dependent variable is inside the logarithm.
\(\quad e^t\)
Linear: It does not contain the dependent variable.
Answer .
Nonlinear terms:
\(yy''\text{,}\) \(y^2\text{,}\) and
\(\ln(y')\text{.}\)
(c)
Select the linear terms in the differential equation:
\begin{equation*}
3t^2 + y \sin(t) = t\sin(y') + e^{ty}
\end{equation*}
\(\quad 3t^2\)
Linear: It does not contain the dependent variable.
\(\quad y \sin(t)\)
Linear: The dependent variable is raised to the power of one.
\(\quad t\sin(y')\)
Not linear: the derivative of the dependent variable appears inside the sine function.
\(\quad e^{ty}\)
Not linear: the dependent variable \(y\) appears in an exponent.
Answer .
Linear terms:
\(3t^2\) and
\(y\sin(t)\text{.}\)
(d)
Which of the following terms is linear?
\(\dfrac{1}{t}y''\)
Correct! \(\dfrac{1}{t}y''\) is linear because it is a function of the independent variable multiplied by the second derivative of the dependent variable.
\(y^3\)
Incorrect. \(y^3\) is nonlinear because the dependent variable is raised to a power other than one.
\(e^t y^2\)
Incorrect. \(e^t y^2\) is nonlinear because the dependent variable is squared.
\(y \cos(y)\)
Incorrect. \(y \cdot \cos(y)\) is nonlinear because it involves the product of the dependent variable and a function of the dependent variable.
Answer .
Linear term:
\(\dfrac{1}{t}y''\text{.}\)
(e)
Click on all of the nonlinear terms in the differential equation.
In this equation, \(y^3\) and \(\ln(y)\) are nonlinear terms.
\(\vphantom{y'}\)
\(y^3 \) \(\ +\ \) \(e^t \dfrac{d^3y}{dt^3} \) \(\ -\ \) \(\ln(y) \) \(\ +\ \) \(t \dfrac{dy}{dt} \) \(\ +\ \) \(\dfrac{d^2y}{dt^2} \) \(\ =\ \) \(0 \)
\(\vphantom{y'}\)
Hint . Answer .
Nonlinear terms:
\(y^3\) and
\(\ln(y)\text{.}\)
5. Identify the Linear & Nonlinear Differential Equations.
(a)
Identify the linearity of the differential equation
\begin{equation*}
y'' + \sin(y) = 17t\text{.}
\end{equation*}
Linear
No, this is nonlinear. Looking carefully at each term, we see:
\begin{gather*}
y'' + \sin(y) = 17t \\
\underset{\text{linear}}{\underline{(1){\color{BurntOrange} y'' }}} +
\underset{\text{nonlinear}}{\underline{\sin({\color{BurntOrange} y})}} =
\underset{\text{linear}}{\underline{17{\color{BurntOrange} t}}}.
\end{gather*}
Since one term is not linear, the entire differential equation is nonlinear.
Nonlinear
Correct! This DE is nonlinear since \(\sin(y)\) is a nonlinear term.
(b)
Identify the linearity of the differential equation
\begin{equation*}
y'' + y' \cos t = 7y\text{.}
\end{equation*}
Linear
Correct! This equation is linear because each term is linear.
Nonlinear
No, this is linear. Looking carefully at each term, we see:
\begin{gather*}
y'' + y' \cos t = 7y \\
\underset{\text{linear}}{\underline{(1){\color{blue} y'' }}} +
\underset{\text{linear}}{\underline{(\cos t){\color{blue} y' }}} =
\underset{\text{linear}}{\underline{7{\color{blue} y}}}.
\end{gather*}
Since every term is linear, this differential equation is linear.
(c)
Identify the linearity of the differential equation
\begin{equation*}
\frac{dy}{dt} + t^2 y = e^t\text{.}
\end{equation*}
Linear
Correct! Since each term is linear, the differential equation is linear.
Nonlinear
Incorrect. Each term is linear since a single dependent variable or its derivative appears to the first power and is not inside a function.
(d)
Identify the linearity of the differential equation
\begin{equation*}
\frac{d^2x}{dt^2} + e^x = 0\text{.}
\end{equation*}
Linear
Incorrect. The term \(e^x\) makes this equation nonlinear, as it involves the exponential function of the dependent variable.
Nonlinear
Correct! The term \(e^x\) introduces nonlinearity into the equation, as it involves the dependent variable \(x\) inside an exponential function.
(e)
Select the linear differential equation.
\(\quad y'' + y^3 = \sin(t)\)
Incorrect. The \(y^3\) term is nonlinear, making the equation nonlinear.
\(\quad y'' + \cos(y) = 0\)
Incorrect. The \(\cos(y)\) term is nonlinear, making the equation nonlinear.
\(\quad y'' + y' + y = 0\)
Correct! All terms in this equation are linear, making it a linear differential equation.
\(\quad y' + y^2 = t\)
Incorrect. The \(y^2\) term is nonlinear, making the equation nonlinear.
Answer .
\(y'' + y' + y = 0\text{.}\)
(f)
Click on all the linear differential equations.
Linear equations only involve the dependent variable and its derivatives to the first power, and they won’t be inside nonlinear functions like sine or multiplied by each other.
\(y'' + \sin(y) = 17t \) \(y'' + \dfrac{y'}{t^2} + y = 17t \) \(y'' + 3y' + 2y = 0 \)
\(\) \(\) \(\)
\(y'' + y^2 = 17t \) \(y'' + \dfrac{y'}{t} + y = 17t \) \(y = y' \)
Hint .
Remember that a linear differential equation contains only linear terms. Four of these equations are linear.
Answer .
Linear equations:
\(y'' + \dfrac{y'}{t^2} + y = 17t\text{,}\) \(y'' + 3y' + 2y = 0\text{,}\) \(y'' + \dfrac{y'}{t} + y = 17t\text{,}\) \(y = y'\text{.}\)
(g)
Click on all the nonlinear differential equations.
\(\dfrac{dx}{dt} = x^2 + t\) \(\dfrac{dy}{dt} + t^2 y = e^t\) \(\dfrac{du}{ds} = \cos(u)\)
\(\) \(\) \(\)
\(\dfrac{d^2w}{dr^2} + 4w = 0 \) \(\dfrac{dp}{dq} = p^2 q \) \(\dfrac{dv}{dz} + v = z^3 \)
\(\) \(\) \(\)
\(\dfrac{dh}{dx} = \sqrt{h} \) \(\dfrac{dT}{dt} + 3T = 20 \) \(y' + (y')^2 = x \)
Hint .
First, identify the dependent variable, then carefully examine each term to determine whether it is nonlinear.
Answer .
Nonlinear equations:
\(\dfrac{dx}{dt} = x^2 + t\text{,}\) \(\dfrac{du}{ds} = \cos(u)\text{,}\) \(\dfrac{dp}{dq} = p^2 q\text{,}\) \(\dfrac{dh}{dx} = \sqrt{h}\text{,}\) \(y' + (y')^2 = x\text{.}\)
✍🏻 Problems.
Complete the following exercises.
6. Determine the Dependent Variable & Order.
7. Determine Whether the Differential Equation Is Linear.
For each differential equation, identify the dependent variable and determine if it is linear.
Differential Equation
Dependent Variable?
Linear?
(a)
\(\dfrac{d^2u}{dr^2} + \dfrac{du}{dr} + u = \cos(r+u) \) \(\ r\ \) \(\ u\ \) yes no
(b)
\(x \dfrac{d^3y}{dx^3} - \left( \dfrac{dy}{dx} \right)^4 + y = 0 \) \(\ x\ \) \(\ y\ \) yes no
(c)
\(\vphantom{\dfrac11} t^5 x^{(4)} - t^3 x'' + 6x = 0 \) \(\ x\ \) \(\ t\ \) yes no
(d)
\(\dfrac{d^2x}{dy^2} = \sqrt{1 + \dfrac{dx}{dy}} \) \(\ x\ \) \(\ y\ \) yes no
(e)
\(\dfrac{d^2R}{dt^2} = -\dfrac{k}{R^2}\) \(\ R\ \) \(\ t\ \) yes no
(f)
\(\vphantom{\dfrac11} (\sin \theta)y''' - (\cos \theta)y' = 2\) \(\ \theta\ \) \(\ y\ \) yes no
Answer .
(a)
\(u\text{,}\) nonlinear; (b)
\(y\text{,}\) nonlinear; (c)
\(x\text{,}\) linear; (d)
\(x\text{,}\) nonlinear; (e)
\(R\text{,}\) nonlinear; (f)
\(y\text{,}\) linear.
8. Classify Each Differential Equation.
For each differential equation, determine the following:
the variable that you are solving for,
the order of the differential equation,
the linear terms, and
the linearity of the equation.
(a)
\(\quad \dfrac{d^2u}{dr^2} + \dfrac{du}{dr} + u = \cos(r)\)
Select the Correct Answer
(a)
Solves for:
\(r\) \(\quad\) \(u\)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\dfrac{d^2u}{dr^2}\) \(\quad\) \(\dfrac{du}{dr}\) \(\quad\) \(u\) \(\quad\) \(\cos(r)\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(u\text{,}\) order 2, linear terms only, linear.
(b)
\(\quad (1 - x)y'' - 4xy' + 5y = \cos x\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\((1 - x)y''\) \(\quad\) \(-4xy'\) \(\quad\) \(5y\) \(\quad\) \(\cos x\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 2, linear terms only, linear.
(c)
\(\quad x \dfrac{d^3y}{dx^3} - \left( \dfrac{dy}{dx} \right)^4 + y = 0\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(x \dfrac{d^3y}{dx^3}\) \(\quad\) \(-\left( \dfrac{dy}{dx} \right)^4\) \(\quad\) \(y\) \(\quad\) \(0\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 3, nonlinear because of
\(\left(\dfrac{dy}{dx}\right)^4\text{.}\)
(d)
\(\quad t^5 y^{(4)} - t^3 y'' = 6y\)
Select the Correct Answer
(a)
Solves for:
\(\ t\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(t^5 y^{(4)}\) \(\quad\) \(t^3 y''\) \(\quad\) \(6y\) \(\quad\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 4, linear.
(e)
\(\quad \dfrac{d^2x}{dr^2} = \sqrt{1 + \left( \dfrac{dx}{dr} \right)^2}\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ r\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\dfrac{d^2x}{dr^2}\) \(\quad\) \(\sqrt{1 + \left( \dfrac{dx}{dr} \right)^2}\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(x\text{,}\) order 2, nonlinear because of the square root of
\(1 + \left(\dfrac{dx}{dr}\right)^2\text{.}\)
(f)
\(\quad \dfrac{d^2R}{dt^2} = -\dfrac{k}{R}\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ R\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\dfrac{d^2R}{dt^2}\) \(\quad\) \(-\dfrac{k}{R}\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(R\text{,}\) order 2, nonlinear because of
\(\dfrac{k}{R}\text{.}\)
(g)
\(\quad (\sin \theta)y''' - (\cos \theta)y' = 2\)
Select the Correct Answer
(a)
Solves for:
\(\ y\ \) \(\quad\) \(\ \theta\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\sin \theta y'''\) \(\quad\) \(-\cos \theta y'\) \(\quad\) \(2\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 3, linear.
(h)
\(\quad y\dfrac{dy}{dx} + 4y = x^6e^x\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(y\dfrac{dy}{dx}\) \(\quad\) \(4y\) \(\quad\) \(x^6e^x\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 1, nonlinear because of
\(y\dfrac{dy}{dx}\text{.}\)
(i)
\(\quad \sin(x)\dfrac{dy}{dx} + 3y = 0\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\sin(x)\dfrac{dy}{dx}\) \(\quad\) \(3y\) \(\quad\) \(0\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(y\text{,}\) order 1, linear.
(j)
\(\quad \dfrac{dP}{dt}+2tP = P + 4t -2\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ P\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\dfrac{dP}{dt}\) \(\quad\) \(2tP\) \(\quad\) \(P\) \(\quad\) \(4t-2\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(P\text{,}\) order 1, linear.
(k)
\(\quad x''' = x^2 - 3x'\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ u\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(x^2\) \(\quad\) \(-3x'\) \(\quad\) \(x'''\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(x\text{,}\) order 3, nonlinear because of
\(x^2\text{.}\)
(l)
\(\quad r''' + p^2 r^{(5)} = r\ln(p)\)
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ r\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(r\ln(p)\) \(\quad\) \(p^2 r^{(5)}\) \(\quad\) \(r'''\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
Answer .
\(r\text{,}\) order 5, linear.
9. Determine the Linearity of Each Term.
Determine the linearity of each term in 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*}
Answer .
Linear:
\(e^{t}y^{(7)}\text{,}\) \(-y'\sin t\text{,}\) and
\(\dfrac{3}{t}\text{.}\) Nonlinear:
\((t+1)y'y'''\text{,}\) \(-t\ln y''\text{,}\) \(-\tan y\text{,}\) and
\(\dfrac{4}{y}\text{.}\)
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