Consider the solution of the differential equation y\' = -4y passing through y(0

Question

Consider the solution of the differential equation y' = -4y passing through y(0) = 1. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1). Use Euler's method with step size Delta x = 0.2 to estimate the solution at x = 0.2,0.4, ... ,1, using these to fill in the following table. (Be sure not to round your answers at each step!) Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? Over under Check that y = e-4x is a solution to y' = -4y with y(0) = 1.

Explanation / Answer

B)y(0) = 1 y' = -4y y(0.2) = y(0)+(0.2*(-1)*4*y(0)) = 0.200 y(0.4) = y(0.2)+(0.2*(-1)*4*y(0.2)) = 0.040 y(0.6) = y(0.4)+(0.2*(-1)*4*y(0.4)) = 0.008 y(0.8) = y(0.6)+(0.2*(-1)*4*y(0.6)) = 0.002 y(1.0) = y(0.8)+(0.2*(-1)*4*y(0.8)) = 0.00032 C)Over D)y=e^(-4x) y' = -4e^(-4x) = -4y So it is a solution.

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