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Monday March 23
Last week we solved differential equations algebraically by separating variables and integrating both sides. Unfortunately, it is not always possible to solve algebraically. In this case, a graphical approach is an option.

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 intial condition when x = 1, y=1. The curve in black is for the intial condition y = 3 when x = 0.

For additional help on slope-fields and differential equations click here

Tuesday Mar 24th
AP problems today. 1997 AB 6 Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, where t is non- negative. After her parachute opens, her velocity satisfies the differential equation dv/dt = -2v - 32, with initial condition v(0) = -50. a) Use separation of variables to find an expression for v in terms of t, where t is measured in seconds. b) Terminal velocity is defined as.

Find the terminal velocity of the skydiver to the nearest foot per second. c)It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed? 1998 BC 4 Consider the differential equation dy/dx = (xy)/2 a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

c) Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = .5. Use your solution to find f(0.2)

2000 AB 6 Consider the differential equation dy/dx = (3x²)/(e^2y).

a) Find a solution y = f(x) to the differential equation satisfying f(0) = .5

2004 AB 6 Consider the differential equation dy/dx = x²(y - 1). On the axes provided, sketch a slope-field for the given differential equation



b) While the slope-filed 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. c) Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 3.
 * The slopes will be positive for all points where x is non-zero and y > 1.**

Wednesday Mar 25
Today we reviewed all the work on differential equations and slope-fields using multiple choice questions from the D&S books. We were also given the option of a worksheet which is attached below. Practice Problems Problem Set Study hard for the test tomorrow!