Solved: Electric Circuits The diagram shows a simple electric circuit consisting of a

In Exercises 18, verify the solution of the differential equation.y
Ce y
4y

In Exercises 18, verify the solution of the differential equation.3y  4y
ex y

In Exercises 18, verify the solution of the differential equation.y
2xy
x2 y2 x 2

In Exercises 18, verify the solution of the differential equation.dx
xy y2 1 y2

In Exercises 18, verify the solution of the differential equation.y
C1 cos x  C2 sin x y

In Exercises 18, verify the solution of the differential equation.
C1e y  2y  2y
0 x cos x  C2ex sin x y

In Exercises 18, verify the solution of the differential equation.y
cos x ln y  y
tan x sec x  tan x

In Exercises 18, verify the solution of the differential equation.y  2y
2ex y

In Exercises 912, verify the particular solution of the differential equation.

In Exercises 912, verify the particular solution of the differential equation.

In Exercises 912, verify the particular solution of the differential equation.

In Exercises 912, verify the particular solution of the differential equation.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1318, determine whether the function is a solution of the differential equationy4  16y 0.

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . yy
x2

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . yy
x2 ex

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . yy
x2
2  ex

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . y
sin x

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . yy
ln x

In Exercises 1924, determine whether the function is a solution of the differential equation xy
 2y x3ex . yy
x2ex 5x2

In Exercises 2528, some of the curves corresponding to different values of in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

In Exercises 2528, some of the curves corresponding to different values of in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

In Exercises 2528, some of the curves corresponding to different values of in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

In Exercises 2528, some of the curves corresponding to different values of in the general solution of the differential equation are given. Find the particular solution that passes through the point shown on the graph.

In Exercises 29 and 30, the general solution of the differential equation is given. Use a graphing utility to graph the particular solutions for the given values of C

In Exercises 29 and 30, the general solution of the differential equation is given. Use a graphing utility to graph the particular solutions for the given values of C

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 3136, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

In Exercises 37 48, use integration to find a general solution of the differential equation.y dx
3x2

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x3 4x dy

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x 1  x2

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
ex 1  ex

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x 2 x

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x cos x2

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
sin 2x

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
tan2 x

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x x 3

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
x 5 x

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
xex2

In Exercises 37 48, use integration to find a general solution of the differential equation.dy dx
5ex2 d

Slope Fields In Exercises 4952, a differential equation and its slope field are given. Determine the slopes (if possible) in the slope field at the points given in the table.dy dx
x y

Slope Fields In Exercises 4952, a differential equation and its slope field are given. Determine the slopes (if possible) in the slope field at the points given in the table.dy dx
x y

Slope Fields In Exercises 4952, a differential equation and its slope field are given. Determine the slopes (if possible) in the slope field at the points given in the table.dy dx
x cos y 8

Slope Fields In Exercises 4952, a differential equation and its slope field are given. Determine the slopes (if possible) in the slope field at the points given in the table.dy dx
tan y 6 

In Exercises 5356, match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]dy dx
cos
2x

In Exercises 5356, match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]dy dx
1 2 sin x

In Exercises 5356, match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]dy dx
e2x

In Exercises 5356, match the differential equation with its slope field. [The slope fields are labeled (a), (b), (c), and (d).]dy dx
1 x

Slope Fields In Exercises 5760, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and x.

Slope Fields In Exercises 5760, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and x.

Slope Fields In Exercises 5760, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and x.

Slope Fields In Exercises 5760, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and x.

Slope Field Use the slope field for the differential equation where to sketch the graph of the solution that satisfies each given initial condition. Then make a conjecture about the behavior of a particular solution of as To print an enlarged copy of the graph, go to the website www.mathgraphs.com

Slope Field Use the slope field for the differential equation where to sketch the graph of the solution that satisfies each initial condition. Then make a conjecture about the behavior of a particular solution of as To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.dx
0.5y,

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.dy dx
2 y,

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.y dx
0.02y
10 y, y

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.dy dx
0.2x
2 y,

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.dy dx
0.4y
3 x, y

Slope Fields In Exercises 6368, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition.y dx
1 2 ex8 sin y 4 , y

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h
x  y, y
0
2, n
10, h
0.1

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size hy
x  y, y
0
2, n
20, h
0.05

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size hy
3x 2y, y
0
3, n
10, h
0.05 y

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size hy
0.5x
3 y, y
0
1, n
5, h
0.4 y

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size hy
e y
0
1, n
10, h
0.1 x

Eulers Method In Exercises 6974, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size hy
cos x  sin y, y
0
5, n
10, h
0.1

In Exercises 7577, complete the table using the exact solution of the differential equation and two approximations obtained using Eulers Method to approximate the particular solution of the differential equation. Use and and compute each approximation to four decimal places.

In Exercises 7577, complete the table using the exact solution of the differential equation and two approximations obtained using Eulers Method to approximate the particular solution of the differential equation. Use and and compute each approximation to four decimal places.

In Exercises 7577, complete the table using the exact solution of the differential equation and two approximations obtained using Eulers Method to approximate the particular solution of the differential equation. Use and and compute each approximation to four decimal places.

Compare the values of the approximations in Exercises 7577 with the values given by the exact solution. How does the error change as increases?

Temperature At time minutes, the temperature of an object is The temperature of the object is changing at the rate given by the differential equation (a) Use a graphing utility and Eulers Method to approximate the particular solutions of this differential equation at 2, and 3. Use a step size of (A graphing utility program for Eulers Method is available on the website college.hmco.com.) (b) Compare your results with the exact solution y
72  68et2 . t

Temperature Repeat Exercise 79 using a step size of h
0.05. Compare the results.

In your own words, describe the difference between a general solution of a differential equation and a particular solution.

Explain how to interpret a slope field.

Describe how to use Eulers Method to approximate the particular solution of a differential equation.

It is known that is a solution of the differential equation Is it possible to determine or from the information given? If so, find its value.

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Error and Eulers Method The exact solution of the differential equation where is (a) Use a graphing utility to complete the table, where is the exact value of the solution, is the approximate solution using Eulers Method with is the approximate solution using Eulers Method with is the absolute error is the absolute error and is the ratio (b) What can you conclude about the ratio as changes? (c) Predict the absolute error when h
0.05.

Error and Eulers Method Repeat Exercise 89 where the exact solution of the differential equation where y
0
1, is y
x 1  2ex y

Electric Circuits The diagram shows a simple electric circuit consisting of a power source, a resistor, and an inductor. A model of the current in amperes at time is given by the first-order differential equation where is the voltage produced by the power source, is the resistance in ohms and is the inductance in henrys Suppose the electric circuit consists of a 24-V power source, a 12- resistor, and a 4-H inductor. (a) Sketch a slope field for the differential equation. (b) What is the limiting value of the current? Explain

Think About It It is known that is a solution of the differential equation Find the values of k

Think About It It is known that is a solution of the differential equation Find the values of y  16y
0. .

Let be a twice-differentiable real-valued function satisfying where for all real Prove that is bounded.

Prove that if the family of integral curves of the differential equation is cut by the line the tangents at the points of intersection are concurrent.