6
Differential Equations
6.1 DE & Slope
Fields
BC Day 1
Copyright © Cengage Learning. All rights reserved.
1
6.1
Differential Equations
and
Slope Fields
Copyright © Cengage Learning. All rights reserved.
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Objectives
 Use initial conditions to find particular
solutions of differential equations.
 Use slope fields to approximate solutions of
differential equations.
 Use Euler’s Method to approximate solutions
of differential equations.
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General and Particular Solutions
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General and Particular Solutions
A differential equation in x and y is an equation that
involves x, y, and derivatives of y. A differential equation
gives the slope of a solution curve at any point in the plane
in terms o the coordinates of the point.
A function y = f(x) is called a solution of a differential
equation if the equation is satisfied when y and its
derivatives are replaced by f(x) and its derivatives.
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General and Particular Solutions
For example, differentiation and substitution would show
that y = e–2x is a solution of the differential equation
y' + 2y = 0.
Show it.
It can be shown that every solution of the above differential
equation is of the form
y = Ce–2x
General solution of y ' + 2y = 0
where C is any real number.
This solution is called the general solution.
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General and Particular Solutions
The order of a differential equation is determined by the highestorder derivative in the equation.
For instance, y' = 4y is a first-order differential equation.
The second-order differential equation s''(t) = –32 has the
general solution
s(t) = –16t2 + C1t + C2 which contains two arbitrary constants.
Show that s(t) is a general solution of the second order
differential equation.
It can be shown that a differential equation of order n has a
general solution with n arbitrary constants.
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General and Particular Solutions
The term “initial condition” stems from the fact that, often in
problems involving time, the value of the dependent variable
or one of its derivatives is known at the initial time t = 0.
For instance, the second-order differential equation
s''(t) = –32 having the general solution
s(t) = –16t2 + C1t + C2 General solution of
might have the following initial conditions.
s(0) = 80, s'(0) = 64
Initial conditions
s''(t) = –32
In this case, the initial conditions yield the particular solution
s(t) = –16t2 + 64t + 80. Particular solution
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Example 1 – Verifying Solutions
Determine whether the function is a solution of
the differential equation y''– y = 0.
a. y = sin x
b. y = 4e–x
c. y = Cex
Solution:
a. Because y = sin x, y' = cos x, and y'' = –sin x, it follows
that
y'' – y = –sin x – sin x = –2sin x ≠ 0.
So, y = sin x is not a solution.
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Example 1 – Solution
cont’d
b. Because y = 4e–x, y' = –4e–x, and y'' = 4e–x, it follows that
y'' – y = 4e–x – 4e–x= 0.
So, y = 4e–x is a solution.
c. Because y = Cex, y' = Cex, and y'' = Cex, it follows that
y'' – y = Cex – Cex= 0.
So, y = Cex is a solution for any value of C.
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General and Particular Solutions
Geometrically, the general solution of a first-order
differential equation represents a family of curves known as
solution curves, one for each value assigned to the
arbitrary constant.
For instance, you can verify that every function of the form
is a solution of the differential equation xy' + y = 0.
Show it.
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General and Particular Solutions
Figure 6.1 shows four of the solution curves corresponding
to different values of C.
Particular solutions of a differential
equation are obtained from initial
conditions that give the values of
the dependent variable or one of its
derivatives for particular values of
the independent variable.
Figure 6.1
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Integrating to find solutions:
 Use integration to find a general solution of the
differential equation:
dy
 x3  4 x
dx
4
x
2
y   2x  C
4
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You try
 Use integration to find a general solution of the
differential equation:
dy
2
 x cos x
dx
1
2
y  sin x  C
2
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6.1 Slope Fields
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Greg Kelly, Hanford High School, Richland, Washington
Slope Fields
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Slope Fields
Solving a differential equation analytically can be difficult or
even impossible. However, there is a graphical approach
you can use to learn a lot about the solution of a differential
equation.
Consider a differential equation of the form
y' = F(x, y)
Differential equation
where F(x, y) is some expression in x and y.
At each point (x, y) in the xy–plane where F is defined, the
differential equation determines the slope y' = F(x, y) of the
solution at that point.
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Slope Fields
If you draw short line segments with slope F(x, y) at selected
points (x, y) in the domain of F, then these line segments
form a slope field, or a direction field, for the differential
equation y' = F(x, y).
Each line segment has the same slope as the solution curve
through that point.
A slope field shows the general shape of all the solutions and
can be helpful in getting a visual perspective of the directions
of the solutions of a differential equation. Slope fields are
graphical representations of a differential equation which give
us an idea of the shape of the solution curves. The solution
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curves seem to lurk in the slope field.
Slope Fields
A slope field
shows the
general shape
of all solutions
of a differential
equation.
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Sketching a Slope Field
Sketch a slope field for the differential equation y   2 x
by sketching short segments of the derivative at
several points.
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
y  2 x
x y y
Draw a segment
with slope of 2.
Draw a segment
with slope of 0.
Draw a segment
with slope of 4.
0
0
0
0
1
0
0
2
0
0
3
0
1
0
2
1
1
2
2
0
4
-1
0
-2
-2
0
-4
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
y  2 x
If you know an initial condition, such as
(1,-2), you can sketch the curve.
By following the slope field, you get a
rough picture of what the curve looks
like.
In this case, it is a parabola.
Slope fields show the general shape
of all solutions of a differential equation.
We can see that there are several different
parabolas that we can sketch in the slope
field with varying values of C.
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
Slope Fields

Create the slope field for the differential equation
Since dy/dx gives us the slope at any point, we
just need to input the coordinate:
dy x

dx y
y
2
1
x
-2
-1
1
-1
2
At (-2, 2), dy/dx = -2/2 = -1
At (-2, 1), dy/dx = -2/1 = -2
At (-2, 0), dy/dx = -2/0 = undefined
And so on….
This gives us an outline of a
hyperbola
-2
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Slope Fields
Given:
dy
2
 x ( y  1)
dx
Let’s sketch the slope field …
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Slope Fields
Match the correct DE with its graph:
dy
H dx  y  2
1. _____
A
B
dy
 x3
F dx
2. _____
dy
 sin x
3. _____
D dx
C
dy
 y  y  1
B dx
8. _____
G
If points with the same slope are
along horizontal lines, then DE
depends only on y
ii)
Do you know a slope at a particular
point?
iii)
If we have the same slope along
vertical lines, then DE depends only
on x
iv)
Is the slope field sinusoidal?
v)
What x and y values make the slope
0, 1, or undefined?
vi)
dy/dx = a(x ± y) has similar slopes
along a diagonal.
vii)
Can you solve the separable DE? 30
F
dy
 x y
G
dx
6. _____
dy
x

E dx y
7. _____
i)
D
dy
 cos x
C
dx
4. _____
dy
 x2  y2 E
A dx
5. _____
In order to determine a slope field
for a differential equation, we
should consider the following:
H
Hints on matching a differential equation with its slope field
:
dy
is only in terms of x, slopes will be parallel along vertical lines.
dx
dy
If
is only in terms of y, slopes will be parallel along horizontal lines.
dx
 If

 Horizontal slopes occur where y' = 0.
y' = 3y – 6x will have horizontal slopes where y=2x.

Very steep slopes often correspond to a denominator = 0.

dy x  y

dx x  y will have zero slopes where y= -x and vertical slopes where y=x.
 If y' is periodic in x, the slope field will appear periodic.
 If slopes are all positive or negative in any quadrant, look for y'
containing the product xy or a quotient of the two.
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Slope Fields
 Which of the following graphs could be the graph of the
solution of the differential equation whose slope field is
shown?
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Slope Fields

1998 AP Question: Determine the correct differential
equation for the slope field:
A)
dy
 1 x
dx
dy
 x2
dx
dy
C)
 x y
dx
dy x
D)

dx y
B)
dy
E)
 ln y
dx
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BC Homework
Day 1: Pg. 409: 13-15 odds, 19-23 odds,31, 37-47,
53-59, odds on all
Day 2 (Euler’s Lesson) : Pg. 410: 69,71,73 and Slope
Fields Worksheet
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Homework
 Slope Fields Worksheet
 BC add pg. 411 69-73 odd
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