DE-BOOK Summary & Exercises

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 $+,$ $-,$ 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.

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.
dependentindependent

2. True or False.

A differential equation is an equation that involves one or more integrals of an unknown function.
Answer.
Falsederivatives
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?

v^{'} (t) = 9.8

Answer. Answer

The dependent variable in this equation is

v .

4. Identify the Variable.

Identify the dependent and independent variables of the following DEs:

y^{″} + x y^{'} = \sin x, \frac{d P}{d t} = P (P - 10)

Identify the Variables.

For each of the following differential equations, identify the independent variable and the dependent variable.

5.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

independent variable:

x

dependent variable:

y

6.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

independent variable:

x

dependent variable:

y

7.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

independent variable:

t

dependent variable:

y

8.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

independent variable:

r

dependent variable:

u

9.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

independent variable:

x

dependent variable:

y

10.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R^{2}}

Answer. Answer

independent variable:

t

dependent variable:

R

11.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

independent variable:

θ

dependent variable:

y

12.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. 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.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

dependent variable:

y

2nd order

14.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

dependent variable:

y

3rd order

15.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

dependent variable:

y

4th order

16.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

dependent variable:

u

2nd order

17.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

dependent variable:

y

2nd order

18.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R}

Answer. Answer

dependent variable:

R

2nd order

19.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

dependent variable:

y

3rd order

20.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. Answer

dependent variable:

x

2nd order

21.

y \frac{d y}{d x} - 4 y = x^{6} e^{x}

Answer. Answer

dependent variable:

y

1st order

22.

\sin (x) \frac{d y}{d x} - 3 y = 0

Answer. Answer

dependent variable:

x

2nd order

23.

\frac{d P}{d t} + 2 t P = P + 4 t - 2

Answer. Answer

dependent variable:

P

1st order

24.

\frac{d x}{d y} = x^{2} - 3 x

Answer. 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.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

independent variable:

x

dependent variable:

y

2nd order
linear

26.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

independent variable:

x

dependent variable:

y

3rd order
nonlinear, term:

{(\frac{d y}{d x})}^{4}

27.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

independent variable:

t

dependent variable:

y

4th order
linear

28.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

independent variable:

r

dependent variable:

u

2nd order
nonlinear, term:

\cos (r + u)

29.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

independent variable:

x

dependent variable:

y

2nd order
nonlinear, term:

{(\frac{d y}{d x})}^{2}

30.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R}

Answer. Answer

independent variable:

t

dependent variable:

R

2nd order
nonlinear, term:

\frac{k}{R}

\frac{1}{R}

31.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

independent variable:

θ

dependent variable:

y

3rd order
linear

32.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. Answer

independent variable:

t

dependent variable:

x

2nd order
nonlinear, term:

{\dot{x}}^{2}

33.

y \frac{d y}{d x} - 4 y = x^{6} e^{x}

Answer. Answer

independent variable:

x

dependent variable:

y

1st order
nonlinear, term:

y \frac{d y}{d x}

34.

\sin (x) \frac{d y}{d x} - 3 y = 0

Answer. Answer

independent variable:

t

dependent variable:

x

2nd order
linear

35.

\frac{d P}{d t} + 2 t P = P + 4 t - 2

Answer. Answer

independent variable:

t

dependent variable:

P

1st order
linear

36.

\frac{d x}{d y} = x^{2} - 3 x

Answer. Answer

independent variable:

y

dependent variable:

x

1st order
nonlinear, term:

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.
dependentindependent

2. True or False.

A differential equation is an equation that involves one or more integrals of an unknown function.
Answer.
Falsederivatives
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?

v^{'} (t) = 9.8

Answer. Answer

The dependent variable in this equation is

v .

4. Identify the Variable.

Identify the dependent and independent variables of the following DEs:

y^{″} + x y^{'} = \sin x, \frac{d P}{d t} = P (P - 10)

Identify the Variables.

For each of the following differential equations, identify the independent variable and the dependent variable.

5.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

independent variable:

x

dependent variable:

y

6.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

independent variable:

x

dependent variable:

y

7.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

independent variable:

t

dependent variable:

y

8.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

independent variable:

r

dependent variable:

u

9.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

independent variable:

x

dependent variable:

y

10.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R^{2}}

Answer. Answer

independent variable:

t

dependent variable:

R

11.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

independent variable:

θ

dependent variable:

y

12.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. 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.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

dependent variable:

y

2nd order

14.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

dependent variable:

y

3rd order

15.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

dependent variable:

y

4th order

16.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

dependent variable:

u

2nd order

17.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

dependent variable:

y

2nd order

18.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R}

Answer. Answer

dependent variable:

R

2nd order

19.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

dependent variable:

y

3rd order

20.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. Answer

dependent variable:

x

2nd order

21.

y \frac{d y}{d x} - 4 y = x^{6} e^{x}

Answer. Answer

dependent variable:

y

1st order

22.

\sin (x) \frac{d y}{d x} - 3 y = 0

Answer. Answer

dependent variable:

x

2nd order

23.

\frac{d P}{d t} + 2 t P = P + 4 t - 2

Answer. Answer

dependent variable:

P

1st order

24.

\frac{d x}{d y} = x^{2} - 3 x

Answer. Answer

dependent variable:

x

1st order

Variables & Linearity.

25.

(1 - x) y^{″} - 4 x y^{'} + 5 y = \cos x

Answer. Answer

independent variable:

x

dependent variable:

y

2nd order
linear

26.

x \frac{d^{3} y}{d x^{3}} - {(\frac{d y}{d x})}^{4} + y = 0

Answer. Answer

independent variable:

x

dependent variable:

y

3rd order
nonlinear, term:

{(\frac{d y}{d x})}^{4}

27.

t^{5} y^{(4)} - t^{3} y^{″} + 6 y = 0

Answer. Answer

independent variable:

t

dependent variable:

y

4th order
linear

28.

\frac{d^{2} u}{d r^{2}} + \frac{d u}{d r} + u = \cos (r + u)

Answer. Answer

independent variable:

r

dependent variable:

u

2nd order
nonlinear, term:

\cos (r + u)

29.

\frac{d^{2} y}{d x^{2}} = \sqrt{1 + {(\frac{d y}{d x})}^{2}}

Answer. Answer

independent variable:

x

dependent variable:

y

2nd order
nonlinear, term:

{(\frac{d y}{d x})}^{2}

30.

\frac{d^{2} R}{d t^{2}} = - \frac{k}{R}

Answer. Answer

independent variable:

t

dependent variable:

R

2nd order
nonlinear, term:

\frac{k}{R}

\frac{1}{R}

31.

(\sin θ) y^{‴} - (\cos θ) y^{'} = 2

Answer. Answer

independent variable:

θ

dependent variable:

y

3rd order
linear

32.

\ddot{x} - (1 - \frac{{\dot{x}}^{2}}{3}) \dot{x} + x = 0

Answer. Answer

independent variable:

t

dependent variable:

x

2nd order
nonlinear, term:

{\dot{x}}^{2}

33.

y \frac{d y}{d x} - 4 y = x^{6} e^{x}

Answer. Answer

independent variable:

x

dependent variable:

y

1st order
nonlinear, term:

y \frac{d y}{d x}

34.

\sin (x) \frac{d y}{d x} - 3 y = 0

Answer. Answer

independent variable:

t

dependent variable:

x

2nd order
linear

35.

\frac{d P}{d t} + 2 t P = P + 4 t - 2

Answer. Answer

independent variable:

t

dependent variable:

P

1st order
linear

36.

\frac{d x}{d y} = x^{2} - 3 x

Answer. Answer

independent variable:

y

dependent variable:

x

1st order
nonlinear, term:

x^{2}

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