Differential Equations and Slope Fields By: Leslie Cade 1 st period.

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

Given:

Step 1: Separation of variables- make sure you have the same variables together on different sides of the equation.

Step 2: Integrate both sides of the equation

2dyxy

dx

2

2

1

1 1

2

dy xdxy

C x Cy

2

1dy xdx

y

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:

Step 4: Solve for y

21 1

2C x C

y

21 1

2x C

y

2

112

yx C

Example of Particular Solution

Given: f(0)= 3

Step 1: Separate the variables like you would with a general solution equation.


( 1)( 2)dy

x ydx

1( 1)

2dy x dx

y

2

1( 1)

2

1ln 2

2

dy x dxy

y x x C

Step 3: Apply the exponential function to both sides of the equation

Step 4: Solve for y

21ln 2

2y x x C

21

22x x C

y e

2

2

2

1

2

1

2

1

2

2

2

2

x xC

x x

x x

y e e

y Ce

y Ce

Step 5: Plug in f(0)=3 and solve for C

Step 6: Plug “C” into the y equals equation you found in step 4

21(0) 0

23 2

3 2

1

Ce

C

C

21

2 2x x

y e

Try Me!

1.

2. f(0)=2

3. f(0)=7

4.

1

2

dyxy

dx

9dy

ydx

2(4 )dy

ydx

2 0dy

ydx

Try Me Answer #1

1. Step 1: Separate the variables on each side 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

2

2

2

2

1

4

1

4

1

4

1

21 1

2

1 1

2

1ln

4x C

xC

x

dyxy

dx

dy xdxy

dy xdxy

y x C

y e

y e e

y Ce

Try Me Answer #2

2. f(0)=2 Step 1: Separation of variables


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

9

9

9

9(0)

9

9

19

19

ln 9

2

2

2

x C

C x

x

x

dyy

dx

dy dxy

dy dxy

y x C

y e

y e e

y Ce

Ce

C

y e

Try Me Answer #3

3. f(0)=7 Step 1: Separation of variables


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

2

2

2

2

2(0)

2

2(4 )

12

4

12

4

ln 4 2

ln 4 2

4

4

4

4

7 4

7 4

3

3 4

x C

C x

x

x

x

dyy

dx

dy dxy

dy dxy

y x C

y x C

y e

y e e

y Ce

y Ce

Ce

C

C

y e

Try Me Answer #4

2

2

2

2 0

12

12

ln 2x C

C x

x

dyy

dx

dy dxy

dy dxy

y x C

y e

y e e

y Ce

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

4.

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 equation where

a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

0x 2

1dy y

dx x

b. Find the particular solution y= f(x) to the differential equation with initial condition f(2)=0.

c. For the particular solution y=f(x) described in part b, find

2

2

1

1

1

1

1

2

1

2

1 1( )2

1

1 1

1

1ln 1

1

1

1

1

0 1

1

0

Cx

C x

x

x

x

dy y

dx x

dy dxy x

y Cx

y e

y e e

y Ce

y Ce

Ce

C e

y e

x

1 1( )2lim1

1

x

xe

e

lim ( )x

f x

Try Me!

Consider the differential equation

a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

2dy

xdx

Slope = -2

Slope = 2

Slope = 4

FRQ 2004 (FORM B) AB 5

Consider the differential equationa. On the axes provided, sketch a slope field for

the given differential equation at the twelve points indicated.

4 ( 2)dy

x ydx

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.

and

c. Find the particular solution y=f(x) to the given differential equation with the initial condition f(0)=0.

0x 2y

5

5

5

5

5

4

4

5

1

5

1

5

1

5

1(0)

5

1

5

( 2)

1

2

1ln 2

5

2

2

2

0 2

2

2 2

xC

x

x

x

dyx y

dx

dy x dxy

y x C

y e e

y Ce

y Ce

Ce

C

y e

C.

FRQ 2004 AB 6


a. On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

2 ( 1)dy

x ydx

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.

and

c. Find the particular solution y=f(x) to the given differential equation with the initial condition f(0)=3.

1y 0x

3

3

3

3

3

3

2

2

3

1

3

1

3

1

3

1

3

1(0)

3

1

3

( 1)

1

1

1ln 1

3

1

1

1

1

1 3

2

1 2

x C

xC

x

x

x

dyx y

dx

x dx dyy

x C y

e y

e e y

Ce y

Ce y

Ce

C

y e

Try Me!

1. f(0)=2

2.

3.

4. f(1)=-1

2 6 4 0dy

ydx

0ydyxe

dx

2 2(1 )dy

y xdx

2dy x

dx y

Try Me Answers

1. 2.

0

2 6 4 0

(3 2)

1

3 2

1

3 2

ln 3 2

3 2

3 2

3 2

3 2

1 2

3 31 2

2 ( )3 3

8 1

3 38

8 2

3 3

x C

C x

x

x

x

x

dyy

dxdy

ydx

dy dxy

dy dxy

y x C

y e

y e e

y Ce

y Ce

y Ce

Ce

C

C

y e

2

2

2

0

1

21

ln2

1ln

2

y

y

y

y

dyxe

dx

e dy xdx

e dy xdx

e x C

y x C

y x C

3. 4. f(1)=-1 2 2

22

22

1 3

1 3

3

(1 )

1(1 )

1(1 )

1

31

31

13

dyy x

dx

dy x dxy

dy x dxy

y x x C

y x x C

yx x C

2 2

2 2

2 2

2 2

2

2

2

1

2

2

( 1) 2(1)

1 2

3

2 3

2 3

dy x

dx y

ydy xdx

ydy xdx

y x C

y x C

C

C

C

y x

y x

Slope Field Examplex 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


where

a. On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.

1dy y

dx x

0x

b. Find the particular solution y=f(x) to the differential equation with the initial condition f(-1)=1 and state its domain.

1

1 1

1

ln ln 1

1

1

1 1 1

2

2 1

dy y

dx x

dx dyx y

x C y

Cx y

y C x

C

C

y x

The domain is x<0

Review!x y y’=4x/y

-1 -1 4-1 0 Und.-1 1 -40 -1 00 0 Und.0 1 01 -1 -41 0 Und.1 1 4

Review Continued…

Step 1: Separation of variables


Step 3: Solve in terms of y to find the general solution

2 2

2 2

2

1 1

2 2

dyy xdxydy xdx

ydy xdx

y x C

y x C

y x C

Review Continued…

f(1)=0 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

2

2

2

2

1

21

ln2

10 ln (1)

2

11

21

21 1

ln2 2

y

y

y

y

dyxe

dx

e dy xdx

e dy xdx

e x C

y x C

C

C

C

y x

Differential Equations and Slope Fields By: Leslie Cade 1 st period.

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Transcript of Differential Equations and Slope Fields By: Leslie Cade 1 st period.