6.1 Differential Equations and Slope Fields

First, a little review:

Consider:2 3y x

then: 2y x 2y x

2 5y x or

It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

However, when we try to reverse the operation:

Given: 2y x find y

2y x C

We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

If we have some more information we can find C.

Given: and when , find the equation for .2y x y4y 1x

2y x C 24 1 C

3 C2 3y x

This is called an initial value problem. We need the initial values to find the constant.

An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

Example

• Find the solution of the initial value problem:

26 4 ; (2) 10dy

x x ydx

Example

• Find the solution of the initial value problem:

10 ; P(0) 25tdPe

dt

Example

• Find the solution of the initial value problem:

32 100; s=50 when t=0ds

tdt

Example

• Find the solution of the initial value problem:

sin ; graph of y = f(x)

passes through the point (0, 2)

dy

d

Application

• A car starts from rest and accelerates at a rate of -0.6t2 + 4 m/s2 for 0 < t < 12. How long does it take for the car to travel 100m?

Application

• An object is thrown up from a height of 2m at a speed of 10 m/s. Find its highest point and when it hits the ground.

Integrals such as are called definite integrals

because we can find a definite value for the answer.

4 2

1x dx

4 2

1x dx

43

1

1

3x C

3 31 14 1

3 3C C

64 1

3 3C C

63

3 21

The constant always cancels when finding a definite integral, so we leave it out!

Integrals such as are called indefinite integrals

because we can not find a definite value for the answer.

2x dx

2x dx31

3x C

When finding indefinite integrals, we always include the “plus C”.

Indefinite Integrals

• Review the list of indefinite integrals on p. 307

Differential Equations: General Solution

• Finding the general solution of a differential equation means to find the indefinite integral (i.e. the antiderivative)

Find the general solution

3) 5

1) 8

dya xdxdy

b xdx x

Separation of variables

• If a differential equation has two variables it is separable if it is of the form

( ) ( ) where h(y) 0dy

g x h ydx

Example

is NOT separable

dycos is separable

dx

dyx y

dx

xy y x

Separation of variables2

Solve the differential equation dy x

dx y

Separation of variables

Solve the differential equation 2 1dy

x ydx

Separation of variables

2Solve the differential equation

1

dy xy

dx x

Separation of variables

2

Solve the initial value problem

4; (0) 2

dy xy

dx y

Separation of variables

Solve the initial value problem

4; ( 2) 1

3

dy yy

dx x

Initial value problems and differential equations can be illustrated with a slope field.

Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

Slope Field Activity

• Given 1. Find the slope for your point2. Sketch a tangent segment across your point. Now do

the same for the rest of the points3. Are you on an equilibrium solution?4. Find your isocline. Is it vertical, horizontal, slant, etc.5. Sketch a possible solution curve through your point6. Is your point an extremum or point of inflection? Is the

graph of y increasing/decreasing, CU or CD?7. What is the value of d2y/dx2 at your point?

2dy

xdx

Slope Field Activity

• Given 1. Find the slope for your point2. Sketch a tangent segment across your point. Now do

the same for the rest of the points3. Are you on an equilibrium solution?4. Find your isocline. Is it vertical, horizontal, slant, etc.5. Sketch a possible solution curve through your point6. Is your point an extremum or point of inflection? Is the

graph of y increasing/decreasing, CU or CD?7. What is the value of d2y/dx2 at your point?

dyy

dx

Slope Field Activity

• Given 1. Find the slope for your point2. Sketch a tangent segment across your point. Now do

the same for the rest of the points3. Are you on an equilibrium solution?4. Find your isocline. Is it vertical, horizontal, slant, etc.5. Sketch a possible solution curve through your point6. Is your point an extremum or point of inflection? Is the

graph of y increasing/decreasing, CU or CD?7. What is the value of d2y/dx2 at your point?

dy x

dx y

Slope Field Activity

• Given 1. Find the slope for your point2. Sketch a tangent segment across your point. Now do

the same for the rest of the points3. Are you on an equilibrium solution?4. Find your isocline. Is it vertical, horizontal, slant, etc.5. Sketch a possible solution curve through your point6. Is your point an extremum or point of inflection? Is the

graph of y increasing/decreasing, CU or CD?7. What is the value of d2y/dx2 at your point?

dy y

dx x

Slope Field Activity

• Given 1. Find the slope for your point2. Sketch a tangent segment across your point. Now do

the same for the rest of the points3. Are you on an equilibrium solution?4. Find your isocline. Is it vertical, horizontal, slant, etc.5. Sketch a possible solution curve through your point6. Is your point an extremum or point of inflection? Is the

graph of y increasing/decreasing, CU or CD?7. What is the value of d2y/dx2 at your point?

dyx y

dx

Hw: p. 312/7-17odd,31-36,39-42