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6.1 Slope Fields and Euler’s Method

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.

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6.1 Slope Fields and Euler’s Method

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  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 = Cex y’ = Cex and y” = Cex So, y = Cex is a solution.

  3. Finding a particular solution For the differential equation xy’ - 3y = 0, verify that y = Cx3 is a solution, and find the particular solution when x = -3 and y = 2. y’ = 3Cx2 = 0 xy’ - 3y = x(3Cx2) - 3(Cx3) With the initial condition x = -3 and y = 2 y = Cx3 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) m = -1 - 1 = -2 @ (0,1) m = 0 - 1 = -1 @ (1,1) m = 1 - 1 = 0 -1 1

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

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