Differential Equations and Slopefields
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Transcript Differential Equations and Slopefields
Differential Equations and Slope
Fields
By: Leslie Cade
1st period
Differential Equations
• A differential equation is an equation which
involves a function and its derivative
• There are two types of differential equations:
– General solution: is when you solve in terms of y
and there is a constant “C” in the problem
– Particular solution: is when you solve for the
constant “C” and then you plug the “C” into the y
equals equation
Example of a General Solution
Equation
dy
xy 2
dx
Given:
Step 1: Separation of variables- make sure you
have the same variables together on different
sides of the equation.
1
y
2
dy xdx
Step 2: Integrate both sides of the equation
1
y 2 dy xdx
1
1
C x2 C
y
2
1
1 2
C x C
y
2
Step 3: Notice how above there is a constant “C”
added to both sides, but you can combine those
constants on one side of the equation and end
up with: 1 1 2
y
2
x C
Step 4: Solve for y
y
1
1 2
x C
2
Example of Particular Solution
dy
( x 1)( y 2)
dx
Given:
f(0)= 3
Step 1: Separate the variables like you would
with a general solution equation.
1
dy ( x 1)dx
y2
Step 2: Integrate both sides of the equation
1
y 2dy ( x 1)dx
1
ln y 2 x 2 x C
2
ln y 2
1 2
x xC
2
Step 3: Apply the exponential function to both
sides of the equation
y2 e
Step 4: Solve for y
y 2 eC e
y 2 Ce
y Ce
1 2
x x
2
1 2
x x
2
1 2
x x
2
2
1 2
x x C
2
Step 5: Plug in f(0)=3 and solve for C
3 Ce
1 2
(0) 0
2
2
3C2
C 1
Step 6: Plug “C” into the y equals equation you
found in step 4
ye
1 2
x x
2
2
Try Me!
dy
1
1.
xy
dx
2
2. f(0)=2
3. f(0)=7
dy
9y
dx
dy
2(4 y )
dx
4. dy 2 y 0
dx
Try Me Answer #1
1.
dy
1
xy
dx
2
1
1
dy
xdx
y
2
1
1
dy
xdx
y
2
1 2
ln y
x C
4
ye
1 2
x C
4
y eC e
y Ce
1 2
x
4
1 2
x
4
Step 1: Separate the
variables on each side of
the equation
Step 2: Integrate both sides
of the equation
Step 3: Apply the
exponential function to
both sides of the equation
to get y by itself
Step 4: Solve for y to find
the general equation
Try Me Answer #2
2. f(0)=2
dy
9y
dx
1
dy 9 dx
y
1
y dy 9dx
ln y 9 x C
y e9 x C
y eC e9 x
y Ce 9 x
2 Ce 9 ( 0 )
C 2
y 2e 9 x
Step 1: Separation of
variables
Step 2: Integrate both sides
of the equation
Step 3: Apply the
exponential function to
both sides of the equation
Step 4: Plug in values given
and solve for the constant
“C”.
Step 5: Plug in the constant
you found in step 4 into
your y equals equation to
find the particular
equation
Try Me Answer #3
3. f(0)=7
dy
2(4 y )
dx
1
dy 2dx
4 y
1
4 y dy 2dx
ln 4 y 2 x C
ln 4 y 2 x C
4 y e 2 x C
4 y e C e 2 x
4 y Ce 2 x
y Ce 2 x 4
7 Ce 2(0) 4
7C4
C 3
y 3e 2 x 4
Step 1: Separation of
variables
Step 2: Integrate both sides
of the equation
Step 3: Apply the
exponential function to
both sides
Step 4: Solve for constant
“C” given values f(0)=7
Step 5: Plug constant into y
equals equation to find the
particular equation
Try Me Answer #4
4.
dy
2y 0
dx
1
dy 2dx
y
1
y dy 2dx
ln y 2 x C
y e 2 x C
y eC e 2 x
y Ce
2 x
Step 1: Separate variables
Step 2: Integrate both
sides
Step 3: Apply exponential
function to both sides
Step 4: Solve in terms of y
to find the general
equation
Slope Fields
• Slope fields are a plot of short line segments
with slopes f(x,y) and points (x,y) lie on the
rectangular grid plane
• Slope fields are sometimes referred to as
direction fields or vector fields
• The line segments show the trend of how slope
changes at each point
no slope (0)
undefined
FRQ 2008 AB 5
Consider the differential
x0
dy
y 1
equation dx x 2
where
a. On the axes provided, sketch a slope field for
the given differential equation at the nine
points indicated.
b. Find the particular solution y= f(x) to the
differential equation with initial condition
f(2)=0.
dy
y 1
dx
x2
1
y 1dy
1
x
1
C
x
y 1 eC e
y 1 Ce
y Ce
0 Ce
1
x
1
2
dx
1
C
x
ln y 1
y 1 e
2
1
x
1
x
1
1
1
C e 2
y 1 e
x0
(
1 1
)
2 x
c. For the particular solution y=f(x) described in
part b, find lim f ( x)
x
lim1 e
x
1 e
1 1
( )
2 x
Try Me!
dy
2x
dx
Consider the differential equation
a. On the axes provided, sketch a slope field for
the given differential equation at the nine
points indicated. Slope = 2
Slope = -2
Slope = 4
FRQ 2004 (FORM B) AB 5
dy
x 4 ( y 2)
dx
Consider the differential equation
a. On the axes provided, sketch a slope field for
the given differential equation at the twelve
points indicated.
b. While the slope field in part a is drawn at only
twelve points, it is defined at every point in the
xy-plane. Describe all points in the xy-plane
for which the slopes are negative.
x 0 and y 2
c. Find the particular solution y=f(x) to the given
differential equation with the initial condition
f(0)=0.
C.
dy
x 4 ( y 2)
dx
1
4
dy
x
y 2 dx
1 5
ln y 2 x C
5
y 2 eC e
y 2 Ce
y 2 Ce
0 2 Ce
C 2
y 2 2e
1 5
x
5
1 5
x
5
1 5
x
5
1
(0)5
5
1 5
x
5
FRQ 2004 AB 6
dy
x 2 ( y 1)
dx
Consider the differential equation
a. On the axes provided, sketch a slope field for
the given differential equation at the twelve
points indicated.
b. While the slope field in part a is drawn at only
twelve points, it is defined at every point in the
xy-plane. Describe all points in the xy-plane
for which the slopes are positive.
y 1 and x 0
c. Find the particular solution y=f(x) to the given
differential equation with the initial condition
f(0)=3.
dy
x 2 ( y 1)
dx
1
2
x
dx
y 1dy
1 3
x C ln y 1
3
e
1 3
x C
3
1 3
x
C 3
e e
Ce
1 3
x
3
1 Ce
1 Ce
y 1
y 1
y 1
1 3
x
3
y
1 3
(0)
3
3
C2
y 1 2e
1 3
x
3
Try Me!
1. f(0)=2
dy
2 6y 4 0
dx
2.
dy
xe y 0
dx
3.
dy
y 2 (1 x 2 )
dx
4. f(1)=-1
dy
2x
dx
y
Try Me Answers
dy
6y 4 0
dx
dy
(3 y 2)
dx
1
dy dx
3y 2
1
3 y 2dy dx
2
1.
ln 3 y 2 x C
3 y 2 e x C
3y 2 e e
C
x
3 y 2 Ce x
3 y Ce x 2
1
2
Ce x
3
3
1
2
2 ( Ce 0 )
3
3
8
1
C
3
3
C 8
y
y
8 x
2
e
3
3
2.
dy
xe y 0
dx
e y dy xdx
y
e
dy xdx
1 2
e x C
2
1 2
y ln x C
2
y
1 2
y ln x C
2
3. dy
y (1 x )
2
2
dx
1
2
dy
(1
x
)dx
2
y
1
2
dy
(1
x
)dx
y2
1 3
1
y x x C
3
1 3
1
y x xC
3
1
y
1 3
x xC
3
4. f(1)=-1 dy
2x
dx
y
ydy 2 xdx
ydy 2 xdx
1 2
y x2 C
2
y 2 2 x 2 C
(1) 2 2(1) 2 C
1 2 C
C 3
y 2 2 x 2 3
y 2 x 3
Slope Field Example
x
y
y’ = x + y
-1
-1
-2
-1
0
-1
-1
1
0
0
0
0
1
-1
0
1
0
1
1
1
2
FRQ 2006 AB 5
dy 1 y
dx
x
Consider the differential equation
where x 0
a. On the axes provided, sketch a slope field for
the given differential equation at the eight
points indicated.
b. Find the particular solution y=f(x) to the
differential equation with the initial condition
f(-1)=1 and state its domain.
dy 1 y
dx
x
1
1
xdx 1 y dy
ln x C ln 1 y
Cx 1 y
y C x 1
1 C 1 1
C2
y 2 x 1
The domain is x<0
Review!
x
y
y’=4x/y
-1
-1
-1
0
-1
0
1
-1
4
Und.
-4
0
0
0
1
0
1
-1
Und.
0
-4
1
1
0
1
Und.
4
Review Continued…
dy
y
x
dx
ydy xdx
ydy xdx
1 2 1 2
y x C
2
2
y2 x2 C
y x2 C
Step 1: Separation of
variables
Step 2: Integrate both
sides of the equation
Step 3: Solve in terms
of y to find the
general solution
Review Continued…
f(1)=0
dy
xe y
dx
e y dy xdx
e
y
dy xdx
1 2
x C
2
1
y ln x 2 C
2
ey
0 ln
1 2
(1) C
2
1
C
2
1
C
2
1
1
y ln x 2
2
2
1
Step 1: Separate the
variables
Step 2: Integrate both
sides
Step 3: Plug in f(1)=0 to
find the constant “C”
Step 4: Plug the constant
you just found into the y
equals equation
Bibliography
• http://www.math.buffalo.edu/~apeleg/mth30
6g_slope_field_1.gif
©Leslie Cade 2011