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Section 1.5 Exercises
Subsection 💡 Conceptual Quiz
Exercises Exercises
1. True or False.
(a)
An equation that contains an "=" sign and at least one derivative is called a derivative equation.
(b)
The expression
\(z^{(18)}\) is the same as
\(z\) to the power of 18.
(c)
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.
(d)
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 variable.
Correct! The dependent variable is a function of the independent variable.
The independent variable is a function of the dependent variable.
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. Multiple Choice.
(a)
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.
(b)
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.
(c)
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.
Their solutions are always functions.
Incorrect. While the solutions to differential equations are often functions, this is not what makes them unique.
(d)
Which of the following is NOT required for an equation to be classified as a differential equation?
An unknown function.
Incorrect. A differential equation includes an unknown function that 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 equals sign is required for an equation to be classified as a differential equation.
(e)
Which variable in the differential equation,
\begin{equation*}
\dfrac{dP}{ds} + \dfrac{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 \(P\text{,}\) the dependent variable in this equation.
independent variable, \(P\)
Incorrect. We are solving for \(P\text{,}\) but it is not the independent variable.
(f)
Which variable, in the differential equation below, does the solution of this equation depend on?
\begin{equation*}
\dfrac{dP}{ds} + \dfrac{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.
(g)
Identify the free term of the differential equation
\begin{equation*}
w''=3tw\text{.}
\end{equation*}
\(\quad 3tw\)
Incorrect. This term involves the dependent variable \(w\text{,}\) so it is not a free term.
\(\quad 3t\)
Incorrect. This is part of the term \(3tw\) (it multiplies \(w\) ), 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. In the standard linear form, the free (forcing) term can be \(0\text{;}\) here it is \(0\text{.}\)
Subsection 🏋️♂️ Practice Drills
Exercises Exercises
1. Identify the Differential Equations.
(a)
Select all the expressions that are differential equations.
The differential equations are the four expressions with derivatives and an equals sign.
\(\)
\(\dfrac{dy}{dx} + 3y - 1 \) \(x^2 + 2x - 5 = 0 \) \(\sin(x) + \cos(x) = 1 \)
\(\)
\(\dfrac{d^2y}{dx^2} - y = e^x \) \(y + 2x \) \(y = y' \)
\(\)
\(\ln(x) + \dfrac{dy}{dx} = x^2 \) \(\sqrt{x} + 5 = 3x \) \(\dfrac{d^3z}{dt^3} - 4z = \cos(t) \)
\(\)
Hint .
There are only four differential equations in this set.
(b)
Select the differential equation.
\(\quad \dfrac{dy}{dx} + 1 = y\)
Correct! This equation involves a derivative, making it a differential equation.
\(\quad x^2 + 3x = 19\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
\(\quad \sin y + e^x = 0\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
\(\quad y^2 + 5 = 0\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
2. Dependent & Independent Variables.
(a)
Identify the independent variable of the differential equation
\begin{equation*}
(1 - x)y'' - 4xy' + 5y = \cos x\text{.}
\end{equation*}
\(\ x\)
Yes! \(x\) is the independent variable.
\(\ y\)
Incorrect. Review the examples.
\(\ y'\)
Incorrect. Review the examples.
(b)
Identify the dependent variable of the differential equation
\begin{equation*}
\dfrac{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.
3. Identify the Coefficients.
(a)
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.
(b)
Select each of the coefficients in the differential equation below.
The coefficients are the functions of the independent variable multiplying each dependent-variable term.
\(t \) \(\dfrac{d^2y}{dt^2} \) \(\ +\ \) \(t^2 \) \(y^2 \) \(\ -\ \) \(4 \) \(y' \) \(\ =\ \) \(y^{-1} \) \(t \) \(\ +\ \) \(\sin(t) \)
\(\phantom{vertical space hack - Is there a better way?}\)
Hint .
Look for the dependent variable in each term. The coefficient is the constant or function that multiplies the dependent variable.
You have attempted
of
activities on this page.
