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Section 1.8 Summary & Exercises
Summary of the Key Ideas.
Dependent & Independent Variables
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
Linearity of a Differential Equation
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 .
In this book, DE stands for
.
Answer . differential equation
The solution to a differential equation is a
.
Answer . function
The solution to a differential equation is represented by the
variable of the equation.
Answer . dependent
Solving a differential equation means finding the
variable as a function of the
variable.
Answer .
dependent independent
2. True or False .
A differential equation is an equation that involves one or more integrals of an unknown function.
Answer .
False derivatives
The dependent variable is a function of the independent varaible.
Answer . True
The independent variable is a function of the dependent varaible.
Answer . False
An Ordinary Differential Equation (ODE) contains more than one independent variable.
Answer . False
3.
What is the dependent variable in the following differential equation?
\begin{equation*}
v'(t) = 9.8
\end{equation*}
Answer . 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*}
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 . Answer independent variable: \(x\) \(y\)
6. \(\displaystyle x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \) Answer . Answer independent variable: \(x\) \(y\)
7. \(\displaystyle t^5 y^{(4)} - t^3 y'' + 6y = 0 \) Answer . Answer independent variable: \(t\) \(y\)
8. \(\displaystyle \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \) Answer . Answer independent variable: \(r\) \(u\)
9. \(\displaystyle \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \) Answer . Answer independent variable: \(x\) \(y\)
10. \(\displaystyle \frac{d^2R}{dt^2} = -\frac{k}{R^2} \) Answer . Answer independent variable: \(t\) \(R\)
11. \(\displaystyle (\sin \theta)y''' - (\cos \theta)y' = 2 \) Answer . Answer independent variable: \(\theta\) \(y\)
12. \(\displaystyle \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \) Answer . Answer independent variable: \(t\) \(x\)
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 . Answer dependent variable: \(y\)
14. \(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0\) Answer . Answer dependent variable: \(y\)
15. \(\ds t^5 y^{(4)} - t^3 y'' + 6y = 0\) Answer . Answer dependent variable: \(y\)
16. \(\ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u)\) Answer . Answer dependent variable: \(u\)
17. \(\ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2}\) Answer . Answer dependent variable: \(y\)
18. \(\ds \frac{d^2R}{dt^2} = -\frac{k}{R}\) Answer . Answer dependent variable: \(R\)
19. \(\ds (\sin \theta)y''' - (\cos \theta)y' = 2\) Answer . Answer dependent variable: \(y\)
20. \(\ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0\) Answer . Answer dependent variable: \(x\)
21. \(\ds y\frac{dy}{dx} - 4y = x^6e^x\) Answer . Answer dependent variable: \(y\)
22. \(\ds \sin(x)\frac{dy}{dx} - 3y = 0\) Answer . Answer dependent variable: \(x\)
23. \(\ds \frac{dP}{dt}+2tP = P + 4t -2\) Answer . Answer dependent variable: \(P\)
24. \(\ds \frac{dx}{dy} =x^2-3x \) Answer . Answer dependent variable: \(x\)
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 . Answer independent variable: \(x\) \(y\)
26. \(\, \ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \) Answer . Answer independent variable: \(x\) \(y\) \(\ds \left( \frac{dy}{dx} \right)^4\)
27. \(\, \ds t^5 y^{(4)} - t^3 y'' + 6y = 0 \) Answer . Answer independent variable: \(t\) \(y\)
28. \(\, \ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \) Answer . Answer independent variable: \(r\) \(u\) \(\ds \cos(r+u)\)
29. \(\, \ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \) Answer . Answer independent variable: \(x\) \(y\) \(\ds \left(\ds \frac{dy}{dx} \right)^2\)
30. \(\, \ds \frac{d^2R}{dt^2} = -\frac{k}{R} \) Answer . Answer independent variable: \(t\) \(R\) \(\ds \frac{k}{R}\) or \(\ds \frac{1}{R}\)
31. \(\, \ds (\sin \theta)y''' - (\cos \theta)y' = 2 \) Answer . Answer independent variable: \(\theta\) \(y\)
32. \(\, \ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \) Answer . Answer independent variable: \(t\) \(x\) \(\ds \dot{x}^2\)
33. \(\ds y\frac{dy}{dx} - 4y = x^6e^x\) Answer . Answer independent variable: \(x\) \(y\) \(\ds y\frac{dy}{dx}\)
34. \(\ds \sin(x)\frac{dy}{dx} - 3y = 0\) Answer . Answer independent variable: \(t\) \(x\)
35. \(\ds \frac{dP}{dt}+2tP = P + 4t -2\) Answer . Answer independent variable: \(t\) \(P\)
36. \(\ds \frac{dx}{dy} =x^2-3x \) Answer . Answer independent variable: \(y\) \(x\) \(\ds x^2\)
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 .
In this book, DE stands for
.
Answer . differential equation
The solution to a differential equation is a
.
Answer . function
The solution to a differential equation is represented by the
variable of the equation.
Answer . dependent
Solving a differential equation means finding the
variable as a function of the
variable.
Answer .
dependent independent
2. True or False .
A differential equation is an equation that involves one or more integrals of an unknown function.
Answer .
False derivatives
The dependent variable is a function of the independent varaible.
Answer . True
The independent variable is a function of the dependent varaible.
Answer . False
An Ordinary Differential Equation (ODE) contains more than one independent variable.
Answer . False
3.
What is the dependent variable in the following differential equation?
\begin{equation*}
v'(t) = 9.8
\end{equation*}
Answer . 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*}
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 . Answer independent variable: \(x\) \(y\)
6. \(\displaystyle x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \) Answer . Answer independent variable: \(x\) \(y\)
7. \(\displaystyle t^5 y^{(4)} - t^3 y'' + 6y = 0 \) Answer . Answer independent variable: \(t\) \(y\)
8. \(\displaystyle \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \) Answer . Answer independent variable: \(r\) \(u\)
9. \(\displaystyle \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \) Answer . Answer independent variable: \(x\) \(y\)
10. \(\displaystyle \frac{d^2R}{dt^2} = -\frac{k}{R^2} \) Answer . Answer independent variable: \(t\) \(R\)
11. \(\displaystyle (\sin \theta)y''' - (\cos \theta)y' = 2 \) Answer . Answer independent variable: \(\theta\) \(y\)
12. \(\displaystyle \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \) Answer . Answer independent variable: \(t\) \(x\)
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 . Answer dependent variable: \(y\)
14. \(\ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0\) Answer . Answer dependent variable: \(y\)
15. \(\ds t^5 y^{(4)} - t^3 y'' + 6y = 0\) Answer . Answer dependent variable: \(y\)
16. \(\ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u)\) Answer . Answer dependent variable: \(u\)
17. \(\ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2}\) Answer . Answer dependent variable: \(y\)
18. \(\ds \frac{d^2R}{dt^2} = -\frac{k}{R}\) Answer . Answer dependent variable: \(R\)
19. \(\ds (\sin \theta)y''' - (\cos \theta)y' = 2\) Answer . Answer dependent variable: \(y\)
20. \(\ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0\) Answer . Answer dependent variable: \(x\)
21. \(\ds y\frac{dy}{dx} - 4y = x^6e^x\) Answer . Answer dependent variable: \(y\)
22. \(\ds \sin(x)\frac{dy}{dx} - 3y = 0\) Answer . Answer dependent variable: \(x\)
23. \(\ds \frac{dP}{dt}+2tP = P + 4t -2\) Answer . Answer dependent variable: \(P\)
24. \(\ds \frac{dx}{dy} =x^2-3x \) Answer . Answer dependent variable: \(x\)
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 . Answer independent variable: \(x\) \(y\)
26. \(\, \ds x \frac{d^3y}{dx^3} - \left( \frac{dy}{dx} \right)^4 + y = 0 \) Answer . Answer independent variable: \(x\) \(y\) \(\ds \left( \frac{dy}{dx} \right)^4\)
27. \(\, \ds t^5 y^{(4)} - t^3 y'' + 6y = 0 \) Answer . Answer independent variable: \(t\) \(y\)
28. \(\, \ds \frac{d^2u}{dr^2} + \frac{du}{dr} + u = \cos(r+u) \) Answer . Answer independent variable: \(r\) \(u\) \(\ds \cos(r+u)\)
29. \(\, \ds \frac{d^2y}{dx^2} = \sqrt{1 + \left(\ds \frac{dy}{dx} \right)^2} \) Answer . Answer independent variable: \(x\) \(y\) \(\ds \left(\ds \frac{dy}{dx} \right)^2\)
30. \(\, \ds \frac{d^2R}{dt^2} = -\frac{k}{R} \) Answer . Answer independent variable: \(t\) \(R\) \(\ds \frac{k}{R}\) or \(\ds \frac{1}{R}\)
31. \(\, \ds (\sin \theta)y''' - (\cos \theta)y' = 2 \) Answer . Answer independent variable: \(\theta\) \(y\)
32. \(\, \ds \ddot{x} - \left( 1 - \frac{\dot{x}^2}{3} \right)\dot{x} + x = 0 \) Answer . Answer independent variable: \(t\) \(x\) \(\ds \dot{x}^2\)
33. \(\ds y\frac{dy}{dx} - 4y = x^6e^x\) Answer . Answer independent variable: \(x\) \(y\) \(\ds y\frac{dy}{dx}\)
34. \(\ds \sin(x)\frac{dy}{dx} - 3y = 0\) Answer . Answer independent variable: \(t\) \(x\)
35. \(\ds \frac{dP}{dt}+2tP = P + 4t -2\) Answer . Answer independent variable: \(t\) \(P\)
36. \(\ds \frac{dx}{dy} =x^2-3x \) Answer . Answer independent variable: \(y\) \(x\) \(\ds x^2\)
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