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Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

Geoffrey W. Cox, Ph.D.

Section 1.8 Summary & Exercises

Summary of the Key Ideas.

  • Differential Equation
    • Differential equations are equations that contain derivatives, where the unknowns are functions.
    • They describe how quantities change over time or space.
  • An Analogy
    • Like standard equations, differential equations aim to solve for an unknown, but this unknown is a function rather than a number.
  • Dependent & Independent Variables
    • Differential equations involve both dependent (unknown function) and independent variables.
    • The derivatives in the equation apply to the dependent variable.
  • Terms & Coefficients
    • Terms in a differential equation are separated by \(+\text{,}\) \(-\text{,}\) or \(=\) signs.
    • Coefficients are constants or functions that multiply the dependent variable or its derivatives.
  • Order of a Differential Equation
    • The order of a differential equation is determined by the highest derivative present.
  • Linearity of a Differential Equation
    • A differential equation is linear if it contains only linear combinations of the dependent variable and its derivatives.
    • Otherwise, it is nonlinear.

Exercises Exercises

Conceptual Review.

1. Fill in the blanks.
Fill in each blank with one of the following terms:
dependent, independent, function, or differential equation.
  1. In this book, DE stands for .
    Answer.
    differential equation
  2. The solution to a differential equation is a .
    Answer.
    function
  3. The solution to a differential equation is represented by the variable of the equation.
    Answer.
    dependent
  4. Solving a differential equation means finding the variable as a function of the variable.
    Answer.
    dependent, independent
2. True or False.
  1. A differential equation is an equation that involves one or more integrals of an unknown function.
    Answer.
    False. A differential equation is an equation that involves one or more derivatives of an unknown function, not integrals.
  2. The dependent variable is a function of the independent varaible.
    Answer.
    True.
  3. The independent variable is a function of the dependent varaible.
    Answer.
    False. The dependent variable is the function, which depends on the independent variable.
  4. An Ordinary Differential Equation (ODE) contains more than one independent variable.
    Answer.
    False. An Ordinary Differential Equation (ODE) contains exactly one independent variable. If it contained more than one, it would be a Partial Differential Equation (PDE).
3.
What is the dependent variable in the following differential equation?
\begin{equation*} v'(t) = 9.8 \end{equation*}
Answer.
The dependent variable in this equation is \(v\text{.}\)

4. Identify the Variable.

Identify the dependent and independent variables of the following DEs:
\begin{equation*} y'' + xy' = \sin x, \quad \frac{dP}{dt} = P(P-10) \end{equation*}
Solution.
The first equation has dependent variable \(y\) and independent variable \(x\text{.}\)
The second has dependent variable \(P\) and independent variable \(t\text{.}\)

Identify the Variables.

For each of the following differential equations, identify the independent variable and the dependent variable.
5.
\(\displaystyle (1 - x)y'' - 4xy' + 5y = \cos x \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
6.
\(\displaystyle x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
7.
\(\displaystyle t^5 y^{(4)} - t^3 y'' + 6y = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(y\)
8.
\(\displaystyle \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \)
Answer.
independent variable: \(r\)
dependent variable: \(u\)
9.
\(\displaystyle \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
10.
\(\displaystyle \frac{d^2R}{dt^2} = -\frac{k}{R^2} \)
Answer.
independent variable: \(t\)
dependent variable: \(R\)
11.
\(\displaystyle (\sin \theta)y''' - (\cos \theta)y' = 2 \)
Answer.
independent variable: \(\theta\)
dependent variable: \(y\)
12.
\(\displaystyle \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(x\)

Determine the Order.

For each of the following differential equations, identify the dependent variable and the order.
13.
\(\ds (1 - x)y'' - 4xy' + 5y = \cos x\)
Answer.
dependent variable: \(y\)
2nd order
14.
\(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0\)
Answer.
dependent variable: \(y\)
3rd order
15.
\(\ds t^5 y^{(4)} - t^3 y'' + 6y = 0\)
Answer.
dependent variable: \(y\)
4th order
16.
\(\ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u)\)
Answer.
dependent variable: \(u\)
2nd order
17.
\(\ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2}\)
Answer.
dependent variable: \(y\)
2nd order
18.
\(\ds \frac{d^2R}{dt^2} = -\frac{k}{R}\)
Answer.
dependent variable: \(R\)
2nd order
19.
\(\ds (\sin \theta)y''' - (\cos \theta)y' = 2\)
Answer.
dependent variable: \(y\)
3rd order
20.
\(\ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0\)
Answer.
dependent variable: \(x\)
2nd order
21.
\(\ds y\frac{dy}{dx} - 4y = x^6e^x\)
Answer.
dependent variable: \(y\)
1st order
22.
\(\ds \sin(x)\frac{dy}{dx} - 3y = 0\)
Answer.
dependent variable: \(x\)
2nd order
23.
\(\ds \frac{dP}{dt}+2tP = P + 4t -2\)
Answer.
dependent variable: \(P\)
1st order
24.
\(\ds \frac{dx}{dy} =x^2-3x \)
Answer.
dependent variable: \(x\)
1st order

Variables & Linearity.

For each of the following differential equations, identify the independent and dependent variables, the order, and whether it is linear or nonlinear. If nonlinear, give one term in the expression that breaks the linearity.
25.
\(\, \ds (1 - x)y'' - 4xy' + 5y = \cos x \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
2nd order
linear
26.
\(\, \ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
3rd order
nonlinear, term: \(\ds \left( \frac{dy}{dx} \right)^4\)
27.
\(\, \ds t^5 y^{(4)} - t^3 y'' + 6y = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(y\)
4th order
linear
28.
\(\, \ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \)
Answer.
independent variable: \(r\)
dependent variable: \(u\)
2nd order
nonlinear, term: \(\ds \cos(r+u)\)
29.
\(\, \ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
2nd order
nonlinear, term: \(\ds \left(\ds \frac{dy}{dx} \right)^2\)
30.
\(\, \ds \frac{d^2R}{dt^2} = -\frac{k}{R} \)
Answer.
independent variable: \(t\)
dependent variable: \(R\)
2nd order
nonlinear, term: \(\ds \frac{k}{R}\) or \(\ds \frac{1}{R}\)
31.
\(\, \ds (\sin \theta)y''' - (\cos \theta)y' = 2 \)
Answer.
independent variable: \(\theta\)
dependent variable: \(y\)
3rd order
linear
32.
\(\, \ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(x\)
2nd order
nonlinear, term: \(\ds \dot{x}^2\)
33.
\(\ds y\frac{dy}{dx} - 4y = x^6e^x\)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
1st order
nonlinear, term: \(\ds y\frac{dy}{dx}\)
34.
\(\ds \sin(x)\frac{dy}{dx} - 3y = 0\)
Answer.
independent variable: \(t\)
dependent variable: \(x\)
2nd order
linear
35.
\(\ds \frac{dP}{dt}+2tP = P + 4t -2\)
Answer.
independent variable: \(t\)
dependent variable: \(P\)
1st order
linear
36.
\(\ds \frac{dx}{dy} =x^2-3x \)
Answer.
independent variable: \(y\)
dependent variable: \(x\)
1st order
nonlinear, term: \(\ds x^2\)

Exercises Exercises

Conceptual Review.

1. Fill in the blanks.
Fill in each blank with one of the following terms:
dependent, independent, function, or differential equation.
  1. In this book, DE stands for .
    Answer.
    differential equation
  2. The solution to a differential equation is a .
    Answer.
    function
  3. The solution to a differential equation is represented by the variable of the equation.
    Answer.
    dependent
  4. Solving a differential equation means finding the variable as a function of the variable.
    Answer.
    dependent, independent
2. True or False.
  1. A differential equation is an equation that involves one or more integrals of an unknown function.
    Answer.
    False. A differential equation is an equation that involves one or more derivatives of an unknown function, not integrals.
  2. The dependent variable is a function of the independent varaible.
    Answer.
    True.
  3. The independent variable is a function of the dependent varaible.
    Answer.
    False. The dependent variable is the function, which depends on the independent variable.
  4. An Ordinary Differential Equation (ODE) contains more than one independent variable.
    Answer.
    False. An Ordinary Differential Equation (ODE) contains exactly one independent variable. If it contained more than one, it would be a Partial Differential Equation (PDE).
3.
What is the dependent variable in the following differential equation?
\begin{equation*} v'(t) = 9.8 \end{equation*}
Answer.
The dependent variable in this equation is \(v\text{.}\)

4. Identify the Variable.

Identify the dependent and independent variables of the following DEs:
\begin{equation*} y'' + xy' = \sin x, \quad \frac{dP}{dt} = P(P-10) \end{equation*}
Solution.
The first equation has dependent variable \(y\) and independent variable \(x\text{.}\)
The second has dependent variable \(P\) and independent variable \(t\text{.}\)

Identify the Variables.

For each of the following differential equations, identify the independent variable and the dependent variable.
5.
\(\displaystyle (1 - x)y'' - 4xy' + 5y = \cos x \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
6.
\(\displaystyle x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
7.
\(\displaystyle t^5 y^{(4)} - t^3 y'' + 6y = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(y\)
8.
\(\displaystyle \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \)
Answer.
independent variable: \(r\)
dependent variable: \(u\)
9.
\(\displaystyle \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
10.
\(\displaystyle \frac{d^2R}{dt^2} = -\frac{k}{R^2} \)
Answer.
independent variable: \(t\)
dependent variable: \(R\)
11.
\(\displaystyle (\sin \theta)y''' - (\cos \theta)y' = 2 \)
Answer.
independent variable: \(\theta\)
dependent variable: \(y\)
12.
\(\displaystyle \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(x\)

Determine the Order.

For each of the following differential equations, identify the dependent variable and the order.
13.
\(\ds (1 - x)y'' - 4xy' + 5y = \cos x\)
Answer.
dependent variable: \(y\)
2nd order
14.
\(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0\)
Answer.
dependent variable: \(y\)
3rd order
15.
\(\ds t^5 y^{(4)} - t^3 y'' + 6y = 0\)
Answer.
dependent variable: \(y\)
4th order
16.
\(\ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u)\)
Answer.
dependent variable: \(u\)
2nd order
17.
\(\ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2}\)
Answer.
dependent variable: \(y\)
2nd order
18.
\(\ds \frac{d^2R}{dt^2} = -\frac{k}{R}\)
Answer.
dependent variable: \(R\)
2nd order
19.
\(\ds (\sin \theta)y''' - (\cos \theta)y' = 2\)
Answer.
dependent variable: \(y\)
3rd order
20.
\(\ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0\)
Answer.
dependent variable: \(x\)
2nd order
21.
\(\ds y\frac{dy}{dx} - 4y = x^6e^x\)
Answer.
dependent variable: \(y\)
1st order
22.
\(\ds \sin(x)\frac{dy}{dx} - 3y = 0\)
Answer.
dependent variable: \(x\)
2nd order
23.
\(\ds \frac{dP}{dt}+2tP = P + 4t -2\)
Answer.
dependent variable: \(P\)
1st order
24.
\(\ds \frac{dx}{dy} =x^2-3x \)
Answer.
dependent variable: \(x\)
1st order

Variables & Linearity.

For each of the following differential equations, identify the independent and dependent variables, the order, and whether it is linear or nonlinear. If nonlinear, give one term in the expression that breaks the linearity.
25.
\(\, \ds (1 - x)y'' - 4xy' + 5y = \cos x \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
2nd order
linear
26.
\(\, \ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
3rd order
nonlinear, term: \(\ds \left( \frac{dy}{dx} \right)^4\)
27.
\(\, \ds t^5 y^{(4)} - t^3 y'' + 6y = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(y\)
4th order
linear
28.
\(\, \ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \)
Answer.
independent variable: \(r\)
dependent variable: \(u\)
2nd order
nonlinear, term: \(\ds \cos(r+u)\)
29.
\(\, \ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
2nd order
nonlinear, term: \(\ds \left(\ds \frac{dy}{dx} \right)^2\)
30.
\(\, \ds \frac{d^2R}{dt^2} = -\frac{k}{R} \)
Answer.
independent variable: \(t\)
dependent variable: \(R\)
2nd order
nonlinear, term: \(\ds \frac{k}{R}\) or \(\ds \frac{1}{R}\)
31.
\(\, \ds (\sin \theta)y''' - (\cos \theta)y' = 2 \)
Answer.
independent variable: \(\theta\)
dependent variable: \(y\)
3rd order
linear
32.
\(\, \ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \)
Answer.
independent variable: \(t\)
dependent variable: \(x\)
2nd order
nonlinear, term: \(\ds \dot{x}^2\)
33.
\(\ds y\frac{dy}{dx} - 4y = x^6e^x\)
Answer.
independent variable: \(x\)
dependent variable: \(y\)
1st order
nonlinear, term: \(\ds y\frac{dy}{dx}\)
34.
\(\ds \sin(x)\frac{dy}{dx} - 3y = 0\)
Answer.
independent variable: \(t\)
dependent variable: \(x\)
2nd order
linear
35.
\(\ds \frac{dP}{dt}+2tP = P + 4t -2\)
Answer.
independent variable: \(t\)
dependent variable: \(P\)
1st order
linear
36.
\(\ds \frac{dx}{dy} =x^2-3x \)
Answer.
independent variable: \(y\)
dependent variable: \(x\)
1st order
nonlinear, term: \(\ds x^2\)
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