S2+Week+02

=Differential equations=

Algebraic Solution
Please view the following videos on the topic of differential equations, a topic covered in chapter 7 of the text book. Following each set of videos you should work the problems suggested for practice. media type="custom" key="8020620" media type="custom" key="8020592" media type="custom" key="8020614"

__Assignment 1__
a)Read examples 1 and 2 on page 318.

b) Do problems 1, 5, 9 and 10 of problem set 7.2 starting on page 320. Check your answers in the back of the book.If you have questions, email me at paula.gentry@woodward.edu using your Woodward email.

c) Read examples 1 and 2 on pages 324-325.

d) Do problems 1 and 5 of problem set 7.3 starting on page 328 Check your answers in the back of the book.If you have questions, email me at paula.gentry@woodward.edu using your Woodward email.

e) Print the following worksheet. Work the problems, then email your solutions to me.

Suggestions to get your solutions to me are:

(i) If you have the equations editor for Microsoft Word, you should download the Word document, work the problems within the document, save it, thn return it to me via email. (ii) If you cannot do what is suggested above, print the pdf file, work the problems, scan your solutions, then email them to me.

(iii) If neither (i) or (ii) are possible, you could take photographs of your worked solutions and email the pictures to me.

Graphical Solution
Often differential equations cannot be solved by separating the variables and integrating both sides. If this is the case, we draw a slope field which provides a graphical representation of the possible solution curves. media type="custom" key="8021982" Below you will find another explanation of slope fields. Of course you could also look in your textbook and read section 7-4 starting on page 333.

Consider the differential equation This implies that the slope of tangent lines to the solution curve at any point (x,y) is equal to the opposite of x/y

At (1,1) the slope of the tangent line will be -1; at (2,2) the slope of the tangent line will be -1; in fact, the slope of the tangent line will be -1 at all points where the x and y coordinates are the opposite of each other.

If the x and y coordinates are opposites, e.g. (1, -1), (-2,2) ..., the slope of the tangent lines will be 1.

All points on the x axis have a y coordinate equal to zero. Using the formula for dy/dx, these tangent lines will have undefined slopes implying that the tangent lines will be vertical. [Note: the origin is not included here because the slope of the tangent line will be of indeterminate form.]

All points on the y axis have x coordinate equal to zero. Using the formula for dy/dx, these tangent lines will have zero slope implying that the tangent lines will be horizontal. [Note: the origin is not included here because the slope of the tangent line will be of indeterminate form.].

Very small tangent lines, with appropriate slopes, can be sketched on a grid as shown below.

This is called a SLOPE FIELD.

The slope field suggests that the general solution of the differential equation may be one of a family of concentric circles. Which circle is the particular solution required will depend on the initial condition given.

In the graphic below, the curve sketched in red is for the initial condition when x = 1, y =1. The curve in black is for the initial condition y = 3 when x = 0.



Assignment 2.
(i) Read pages 333 - 336. (ii) Work problems 1, 5, 9. If you have questions, follow the directions given in assignment 1. (iii) Print the following worksheets. Work the problems, then email your solutions to me,following the directions given in assignment 1.



Numerical Solution
Given a differential equation and an initial condition, Euler's method calculates an approximation of a second function value using very short tangent lines. The shorter the tangent lines, the better the approximation. Consider the DE below with an initial condition of y = 2 when x = 1. find an approximation for y when x = 2.



To see a graphical interpretation of Euler's method follow the link below. []

Assignment 3
(i) Read pages 341-343. You will notice a few differences in the method used in the graphic above and the method the author of your textbook suggests. Hopefully you will see the parallels. In short, your book refers to the product of the slope of the tangent line and (x- x 1 ) as dy, the change in the y coordinate, and calculates a new approximation for y by adding dy to the current y value.Both methods are acceptable and are, in fact, equivalent. Of course, it is possible to write a program run Euler's Method for small increments in x; more accurate approximations would result but require more repetitions of th method which would be very tedious by hand.

(ii) Answer the following practice problem. Consider the differential equation dy/dx = 2x - y. Let f(x) be the solution to the differential equation with initial condition f(0) = 1. Use Euler's Method starting at x = 0 with two steps of equal size, to approximate f(-0.4). Show all work. You should arrive at the answer 1.52; if you cannot get this answer, email your questions to me and I will help you.

(iii) Print the following worksheet. Work the problem, then email your solutions to me,following the directions given in assignment 1.

Good Luck!