6.1 Slope Fields and Euler’s Method. Verifying Solutions Determine whether the function is a solution of the Differential equation y” - y = 0 a. y = sin.

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Presentation transcript:

6.1 Slope Fields and Euler’s Method

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.

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

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) m = = (0,1) m = = (1,1) m =1 - 1 = 0

Sketch a slope field for the differential equation y’ = 2x + y that passes through the point (1,1). y|x