1. 6.1 Slope Fields and Euler’s Method

2. Verifying Solutions Determine whether the function is a solution of the Differential equation y” - y = 0 a. y = sin x y’ = cos x and y” = -sin x So, y = sin x is not a solution. b. y = 4e -x y’ = -4e -x and y” = 4e -x So, y = 4e -x is a solution. c. y = Ce x y’ = Ce x and y” = Ce x So, y = Ce x is a solution.

3. Finding a particular solution For the differential equation xy’ - 3y = 0, verify that y = Cx 3 is a solution, and find the particular solution when x = -3 and y = 2. y’ = 3Cx 2 xy’ - 3y = x(3Cx 2 ) - 3(Cx 3 ) = 0 With the initial condition x = -3 and y = 2 y = Cx 3 2 = C(-3) 3 The particular solution is

4. Sketching a Slope Field Sketch a slope field for the differential equation y’ = x - y for the points (-1,1), (0,1), and (1,1). -1 1 @ (-1,1) m =-1 - 1 = -2 @ (0,1) m = 0 - 1 = -1 @ (1,1) m =1 - 1 = 0

5. Sketch a slope field for the differential equation y’ = 2x + y that passes through the point (1,1). y|x-2-1.5-.50.511.52 -2-6-5-4-3-2012 -1.5-5.5-4.5-3.5-2.5-1.5-.5.51.52.5 -5-4-3-20123 0-4-3-201234 12 2 3 4