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Math 2250-4

Week 2 concepts and homework, due September 6.

Recall that problems which are not underlined are good for seeing if you can work with the underlying concepts; that the underlined problems are to be handed in; and that the Friday quiz will be drawn from all of these concepts and from these or related problems.

There will be a Maple/software assignment posted this week. It won't require you to hand in any work on

September 6, but there will be Maple work due on September 13. See our homework page for details.

*1.3 Slope fields and solution curves: understand how the graph of the solution to a first order DE IVP is related to the underlying slope field; the existence-uniqueness theorem for solutions to IVPs.*

*1.3: *

*2*

*, 3, 5, *

*6, *

*10, *

*11*

*, *

*13, 14*

*, *

*23*

*.*

For

**2**

,

**6**

, you may just xerox the book's slope fields. Alternately you may choose to google the applet "dfield", which has dialog boxes that you use to have the software draw these slope fields for you. You can have dfield graph the solution graphs to the desired initial value problems, take screen shots, and print them out to hand in. For

**23**

I recommend dfield.

**w2.1**

) Consider the differential equation in 1.3.6, namely

*y*

#

*x*

=

*x*

K

*y*

C

1.

**a)**

Show that the functions

*y x*

=

*x*

C

*C*

e solve this differential equation.

**b**

) Find the value of

*C*

in the general solution above, so that

*y x*

solves the initial value problem

*y*

#

*x y*

=

K

*x*

K

2 =

*y*

K

C

2.

1

Identify the graph of this solution on your slope field from

**6**

.

1.4:

*2,3,*

*4*

*,9,*

*12*

*,13,19,20,*

*21*

*: solving DE's and IVP's for separable differential equations*

*36*

*, 41, *

*42*

*, 45, *

*46*

*, *

*49*

*, *

*56*

:

*modeling and solving problems with first order separable DEs. Note: The correct answer to 42 is 4.25 billion years, not the 1.25 billion years that appears in the back of the book.*

1.5:

*1*

*, 7, *

*8,*

* 13, *

*24,*

*solving linear DE's and IVP's.*

**w2.2**

) The differential equation in problem

**w2.1**

above is linear. Find the general solution using the integrating factor algorithm for linear differential equations. Verify that you get the same general solution functions

*y x*

as were given to you in part

**w2.1a**

.