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Section Chapter 1 Exercises
Reading Questions α―β
Quick-Answer Questions
1. True or False.
(a) True-or-False.
An equation that contains an "=" sign and at least one derivative is called a derivative equation.
True
Incorrect, derivative equation is not a standard term in mathematics.
False
Correct!
(b) True-or-False.
The expression \(z^{(18)}\) is the same as \(z\) to the power of 18.
True
Incorrect. Please read the note on derivative notation.
False
Correct!
(c) True-or-False.
The order of a differential equation is determined by the number of terms it contains
True
Incorrect. The order is based on the highest derivative, regardless of the number of terms.
False
Correct! The order is determined by the highest derivative, not the number of terms.
(d) True-or-False.
In a differential equation, the dependent variable always has at least one derivative applied to it.
True
Correct! The dependent variable in a differential equation always has a derivative applied to it.
False
Incorrect. By definition, a differential equation involves derivatives of the dependent variable.
(e) True-or-False.
A linear term can contain the dependent variable multiplied by the independent variable.
True
Correct! For example, \(t y\) is a linear term.
False
Incorrect. Carefully review the examples above.
(f) Select All the TRUE statements.
Select All the TRUE statements
For an equation to be a differential equation, it must contain a first-order derivative.
A differential equation must contain a derivative of any order.
The dependent variable is a function of the independent varaible.
Incorrect. Please review the definition of ODEs and PDEs.
The independent variable is a function of the dependent varaible.
The dependent variable is the function, which depends on the independent variable.
An ordinary differential equation (ODE) contains exactly one independent variable.
An ordinary differential equation (ODE) contains exactly one independent variable. If it contained more than one, it would be a partial differential equation (PDE).
2. Differential Equations.
(a) Click-Answer.
Click on all the expressions that are Differential Equations.
\(\dfrac{dy}{dx} + 3y - 1 \) \(x^2 + 2x - 5 = 0 \) \(\sin(x) + \cos(x) = 1 \)
\(\ds \frac{d^2y}{dx^2} - y = e^x \) \(\ds y + 2x \) \(\ds y = y' \)
\(\ds \ln(x) + \frac{dy}{dx} = x^2 \) \(\ds \sqrt{x} + 5 = 3x \) \(\ds \frac{d^3z}{dt^3} - 4z = \cos(t) \)
\(\ds x^2 + y^2 = r^2 \) \(\ds f'(x) + f(x) = 2 \) \(\ds \frac{1}{x} + 3 \)
Hint .
There are only 5 Differential Equations in this set.
(b) Fill-in-the-Blank.
Differential equations differ from algebraic equations in that they contain \(\ul{\qquad}\text{.}\)
solutions
Incorrect. While this statement is generally true, it is not what makes it different from any other equation.
\(y\) variables
Incorrect. Any equation could contain a \(y\) variable.
unknowns
Incorrect. Most equations contain an unknown.
derivatives
Correct! If an equation contains a derivative, it is a differential equation.
(c) Select-the-Best-Answer.
Identify the differential equation.
\(\quad \ds\frac{dy}{dx} + 1 = y\) Correct! This equation involves a derivative, making it a differential equation.
\(\quad x^2 + 3x = 19\) Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
\(\quad \sin y + e^x = 0\) Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
\(\quad y^2 + 5 = 0\) Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
(d) Select-the-best-Answer.
What distinguishes an ordinary differential equation (ODE) from a partial differential equation (PDE)?
The number of variables the unknown function depends on.
Correct! An ODE has derivatives with respect to a single variable, while a PDE involves multiple variables.
The number of derivatives in the equation.
Incorrect. Please review the definition of ODEs and PDEs.
The number of solutions the equation has.
Incorrect. Please review the definition of ODEs and PDEs.
The number of hours it takes to solve the equation.
Incorrect. Please review the definition of ODEs and PDEs.
(e) What makes Differential Equations Unique?
What makes differential equations different from other equations?
They involve derivatives of an unknown function.
Correct! Differential equations are defined by their inclusion of derivatives.
They have many solutions.
Incorrect. While many differential equations can have multiple solutions, this is not what makes them unique.
They involve \(y\) variables.
Incorrect. Any equation could contain \(y\) as a variable.
There solutions are always functions.
Incorrect. While the solutions to differential equations are often functions, this is not what makes them unique.
(f) Select-the-Best-Answer.
Which of the following is NOT required for an equation to be classified as a differential equation?
An unknown function.
Incorrect. A differential equation does include an unknown function, which we are solving for.
An \(x\) -variable.
Correct! An \(x\) -variable is not a requirement for a differential equation.
A derivative.
Incorrect. The presence of at least one derivative is essential to define a differential equation.
An "=" sign.
Incorrect. An equality sign is required for an equation to be classified as a differential equation.
(g) Select-the-Best-Answer.
Which of the following equations is a third-order differential equation?
\(\quad \ds\frac{d^3y}{dx^3} + x^2y = 0\) Correct! The highest derivative here is the third derivative, making it a third-order differential equation.
\(\quad \ds\frac{d^2y}{dx^2} + y' = \sin x\) Incorrect. This is a second-order differential equation.
\(\quad y'' + y' + y = 0\) Incorrect. This is a second-order differential equation.
\(\quad y' + y = x\) Incorrect. This is a first-order differential equation.
(h) Select-the-Best-Answer.
Identify the independent variable of the differential equation
\begin{equation*}
(1 - x)y'' - 4xy' + 5y = \cos x.
\end{equation*}
\(\ x\) Yes! \(x\) is the independent variable.
\(\ y\) Incorrect. Review the examples.
\(\ y'\) Incorrect. Review the examples.
(i) Select-the-Best-Answer.
Identify the dependent variable of the differential equation
\begin{equation*}
\frac{dy}{dx} + 2y = 3x^2
\end{equation*}
\(\ dy/dx\) Incorrect. \(dy/dx\) represents the derivative of the dependent variable with respect to the independent variable.
\(\ x\) Incorrect. The dependent variable is the one being differentiated.
\(\ y\) Correct! \(y\) is the dependent variable in this equation.
(j) Select-the-Best-Answer.
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?
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.
(k) Select-the-Best-Answer.
Which variable, in the differential equation below, does the solution of this equation depend on?
\begin{equation*}
\frac{dP}{ds} + \frac{P}{s^2} = 17s
\end{equation*}
The solution, \(P\text{,}\) depends on the dependent variable, \(s\)
Incorrect. The solution depends on \(s\text{,}\) but \(s\) is not a dependent variable.
The solution, \(P\text{,}\) depends on the independent variable, \(s\)
Yes! the solution, \(P\text{,}\) depends on the independent variable \(s\text{.}\)
The solution, \(s\text{,}\) depends on the dependent variable, \(P\)
Incorrect. \(P\) is the solution, so it does not depend on \(P\text{.}\)
The solution, \(s\text{,}\) depends on the independent variable, \(P\)
Incorrect. The variable \(P\) is not the independent variable.
(l) Fill-in-the-Blank.
Identify the coefficient of \(y'\) in the differential equation
\begin{equation*}
5y'' + 2\cos(t)y' - y = 7
\end{equation*}
\(\quad \cos(t)\) 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.
(m) Click-on-the-Answer.
Click on each of the coefficients in the differential equation below.
\(\phantom{vertical space hack - Is there a better way?}\)
\(t \) \(\ds \frac{d^2y}{dt^2} \) \(\ +\ \) \(\ds t^2 \) \(\ds y^2 \) \(\ -\ \) \(\ds 4 \) \(\ds y' \) \(\ =\ \) \(\ds y^{-1} \) \(\ds t \) \(\ +\ \) \(\ds \sin(t) \) \(\phantom{vertical space hack - Is there a better way?}\)
Hint .
Look for the dependent variable in each term. The coefficient is the constant or function that multiplies the dependent variable.
(n) Select-the-Best-Answer.
Identify the free term in the differential equation
\begin{equation*}
3t^2 y' - 4t^2 = \frac{1}{t} y.
\end{equation*}
\(\quad 3t^2 y'\) 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 \(y\text{.}\)
(o) Select-the-Best-Answer.
Identify the free term of the differential equation
\begin{equation*}
w''=3tw.
\end{equation*}
\(\quad 3tw\) 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.
3. Linear Terms & Linearity.
(a) Select-the-Best-Answer.
Identify the nonlinear terms in the differential equation:
\begin{equation*}
yy'' + y^2 + \ln(y') = e^t
\end{equation*}
\(\quad\ds yy''\) Selected
\(\quad\ds y^2\) Selected
\(\quad\ds \ln(y')\) Selected
\(\quad\ds e^t\) Selected
(b) Select-all-that-Apply.
Select the linear terms in the differential equation:
\begin{equation*}
3t^2 + y \sin(t) = t\sin(y') + e^{ty}
\end{equation*}
\(\quad 3t^2\) Selected
\(\quad y \sin(t)\) Selected
\(\quad t\sin(y')\) Selected
\(\quad e^{ty}\) Selected
(c) Select-the-Best-Answer.
Which of the following terms is linear?
\(\ds \frac{1}{t}y''\) Correct! \(\ds\frac{1}{t}y''\) is linear because it is a function of the independent variable multiplied by the second derivative of the dependent variable.
\(\ds y^3\) Incorrect. \(y^3\) is nonlinear because the dependent variable is raised to a power other than one.
\(\ds e^t y^2\) Incorrect. \(e^t y^2\) is nonlinear because the dependent variable is squared.
\(\ds y \cos(y)\) Incorrect. \(y \cdot \cos(y)\) is nonlinear because it involves the product of the dependent variable and a function of the dependent variable.
(d) Select-the-Best-Answer.
Which term is an example of a nonlinear term?
\(\quad\ds 3\) Incorrect. \(3\) is linear because it is a constant.
\(\quad\ds 3t\) Incorrect. \(3t\) is linear because it is a function of the independent variable only.
\(\quad\ds y^2\) Correct! \(y^2\) is nonlinear because the dependent variable is squared.
\(\quad\ds 2t^2 y\) Incorrect. \(2t^2 y\) is linear because it is a function of the independent variable multiplied by the dependent variable.
(e) Select-the-Best-Answer.
Which term makes the equation \(\ds y''' + 3y' \sin(t) + y^2 = 0\) 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.
(f) Select-the-Best-Answer.
Which of the following describes an example of a nonlinear term?
A dependent variable inside another function.
Correct! This is 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.
(g) Click-on-the-Answer.
Click on all of the linear terms in the differential equation.
\(\phantom{vertical space hack - Is there a better way?}\)
\(\ds \frac{d^2y}{dt^2} \) \(\ +\ \) \(\ds t^2 y \) \(\ +\ \) \(\ds y^2 \) \(\ -\ \) \(\ds \sin(t) y' \) \(\ =\ \) \(\ds 3t \) \(\phantom{vertical space hack - Is there a better way?}\)
(h) Click-on-the-Answer.
Click on all of the nonlinear terms in the differential equation. In this equation, \(y^3\) and \(\ln(y)\) are nonlinear terms.
\(\phantom{vertical space hack - Is there a better way?}\)
\(\ds y^3 \) \(\ +\ \) \(\ds e^t \frac{d^3y}{dt^3} \) \(\ -\ \) \(\ds \ln(y) \) \(\ +\ \) \(\ds t \frac{dy}{dt} \) \(\ +\ \) \(\ds \frac{d^2y}{dt^2} \) \(\ =\ \) \(\ds 0 \) \(\phantom{vertical space hack - Is there a better way?}\)
(i) Select-the-Best-Answer.
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.
(j) Select-the-Best-Answer.
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.
(k) Select-the-Best-Answer.
Identify the linearity of the differential equation
\begin{equation*}
\frac{dy}{dt} + t^2 y = e^t.
\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.
(l) Select-the-Best-Answer.
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.
(m) Select-the-Best-Answer.
Select the linear differential equation.
\(\quad\ds y'' + y^3 = \sin(t)\) Incorrect. The \(y^3\) term is nonlinear, making the equation nonlinear.
\(\quad\ds y'' + \cos(y) = 0\) Incorrect. The \(\cos(y)\) term is nonlinear, making the equation nonlinear.
\(\quad\ds y'' + y' + y = 0\) Correct! All terms are linear in this equation, making it a linear differential equation.
\(\quad\ds y' + y^2 = t\) Incorrect. The \(y^2\) term is nonlinear, making the equation nonlinear.
(n) Click-on-the-Answer.
Click-on all the linear differential equations. Linear equations only involve the dependent variable and its derivatives to the first power, and they wonβt be inside nonlinear functions like sine or multiplied by each other.
\(\ds y'' + \sin(y) = 17t \) \(\ds y'' + \frac{y'}{t^2} + y = 17t \) \(\ds y'' + 3y' + 2y = 0 \)
\(\)
\(\ds y'' + y^2 = 17t \) \(\ds y'' + \frac{y'}{t} + y = 17t \) \(\ds y = y' \)
Hint .
Remember that a linear differential equation contains only linear terms. Four of these equations are linear.
(o) Click-on-the-Answer.
Click-on all the nonlinear differential equations Nonlinear equations often have terms where the dependent variable or its derivatives are raised to powers other than one, or are inside functions like sine, logarithms, or are multiplied by each other.
\(\ds \frac{dx}{ds} = x^2 - 4 \) \(\ds \frac{d^2u}{dz^2} - 5 \frac{du}{dz} + 6u = 0 \) \(\ds \frac{dp}{d\tau} + \sin(p) = \tau^2 \)
\(\)
\(\ds \frac{dw}{dv} + 2vw = \cos(v) \) \(\ds \frac{dr}{d\theta} + r^3 = \theta \) \(\ds \frac{dN}{dt} = -N \)
\(\)
\(\ds \frac{dm}{dq} = m^3 - q^2 \) \(\ds \frac{dz}{dt} + z\frac{dz}{dt} = t^3 \) \(\ds \frac{dy}{dx} = y \ln(y) \)
Hint .
First identify the dependent variable, then carefully look at each term to determine if it is nonlinear.
(p) Click-On Answer.
For each differential equation, identify the dependent variable and determine if it is linear.
(a)
\(\ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \) \(\ r\ \) \(\ u\ \) yes no
(b)
\(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \) \(\ x\ \) \(\ y\ \) yes no
(c)
\(\ds\vphantom{\frac11} t^5 x^{(4)} - t^3 x'' + 6x = 0 \) \(\ x\ \) \(\ t\ \) yes no
(d)
\(\ds \frac{d^2x}{dy^2} = \sqrt{1 + \frac{dx}{dy}} \) \(\ x\ \) \(\ y\ \) yes no
(e)
\(\ds\frac{d^2R}{dt^2} = -\frac{k}{R^2}\) \(\ R\ \) \(\ t\ \) yes no
(f)
\(\ds\vphantom{\frac11} (\sin \theta)y''' - (\cos \theta)y' = 2\) \(\ \theta\ \) \(\ y\ \) yes no
\(\phantom{Extra Vertical Space}\)
4. Fill-In & Drag-N-Drop.
(b) Determine the Dependent Variable & Order.
(f) Drag-and-Drop.
Consider the differential equation
\begin{equation*}
y'' + y' \cos t = 7e^y.
\end{equation*}
Drag each expression (left), to the appropriate label (right).
\(y\) Dependent Variable
\(t\) Independent Variable
\(y' \cos t\) Linear Term
\(7e^y\) Non-Linear Term
\(2\) Order of the DE
\(1\) Coefficient of \(y''\)
\(\cos t\) Coefficient of \(y'\)
Exercises βπ» Classification
Classification Drills.
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.
1. \(\dfrac{d^2u}{dr^2} + \dfrac{du}{dr} + u = \cos(r)\) .
Select the Correct Answer
(a)
Solves for:
\(r\) \(\quad\) \(u\)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\ds\frac{d^2u}{dr^2}\) \(\quad\) \(\ds\frac{du}{dr}\) \(\quad\) \(u\) \(\quad\) \(\cos(r)\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
2. \(\ds (1 - x)y'' - 4xy' + 5y = \cos x\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\ds(1 - x)y''\) \(\quad\) \(\ds -4xy'\) \(\quad\) \(5y\) \(\quad\) \(\cos x\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
3. \(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\ds x \frac{d^3y}{dx^3}\) \(\quad\) \(\ds -\left( \frac{dy}{dx} \right)^4\) \(\quad\) \(y\) \(\quad\) \(0\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
4. \(\ds t^5 y^{(4)} - t^3 y'' = 6y\) .
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
5. \(\ds \frac{d^2x}{dr^2} = \sqrt{1 + \left(\ds \frac{dx}{dr} \right)^2}\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ r\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\ds\frac{d^2x}{dr^2}\) \(\quad\) \(\sqrt{1 + \left(\ds \frac{dx}{dr} \right)^2}\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
6. \(\ds \frac{d^2R}{dt^2} = -\frac{k}{R}\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ R\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\ds\frac{d^2R}{dt^2}\) \(\quad\) \(\ds -\frac{k}{R}\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
7. \(\ds (\sin \theta)y''' - (\cos \theta)y' = 2\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ \theta\ \)
(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
8. \(\ds y\frac{dy}{dx} + 4y = x^6e^x\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(y\frac{dy}{dx}\) \(\quad\) \(4y\) \(\quad\) \(x^6e^x\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
9. \(\ds \sin(x)\frac{dy}{dx} + 3y = 0\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ y\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\sin(x)\frac{dy}{dx}\) \(\quad\) \(3y\) \(\quad\) \(0\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
10. \(\ds \frac{dP}{dt}+2tP = P + 4t -2\) .
Select the Correct Answer
(a)
Solves for:
\(\ x\ \) \(\quad\) \(\ P\ \)
(b)
Order:
1st \(\quad\) 2nd \(\quad\) 3rd \(\quad\) 4th \(\quad\) 5th
(c)
Linear terms:
\(\frac{dP}{dt}\) \(\quad\) \(2tP\) \(\quad\) \(P\) \(\quad\) \(4t-2\)
(d)
Linearity:
Linear \(\quad\) Nonlinear
11. \(\ds x''' = x^2 - 3x'\) .
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
12. \(\ds r''' + p^2 r^{(5)} = r\ln(p)\) .
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
13. 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*}
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