Consider the differential equation dy / dx = 4x, with initial condition y(0) = 4
ID: 3212747 • Letter: C
Question
Consider the differential equation dy / dx = 4x, with initial condition y(0) = 4. Use Euler's method with two steps to estimate y when x = 1: y(1) (Be sure not to round your calculations at each step!) Now use four steps: y(1) (Be sure not to round your calculations at each step!) What is the solution to this differential equation (with the given initial condition)? Y = What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps = Magnitude of error in Euler with 4 steps = By what factor should the error in these approximations change (that is: the error with two steps should be what number times the error with four)? Factor = (How close to this is the result you obtained above?)Explanation / Answer
A)y' = 4x y(0) = 4 y(0.5) = y(0)+(0.5*4*0.5) = 4+(0.5*4*0.5) = 5 y(1) = y(0.5)+(0.5*4*1) = 5+(0.5*4*1) = 7 B)y' = 4x y(0) = 4 y(0.25) = y(0)+(0.25*4*0.25) = 4+(0.25*4*0.25) = 4.250 y(0.5) = y(0.25)+(0.25*4*0.5) = 4.250+(0.25*4*0.5) = 4.750 y(0.75) = y(0.5)+(0.25*4*0.75) = 4.75+(0.25*4*0.75) = 5.500 y(1) = y(0.75)+(0.25*4*1) = 5.5+1 = 6.5 C)dy/dx = 4x =>dy=4xdx Integrating we get, y = 2x^2+c y at x = 0 is 4. =>c = 4 =>y = 2x^2+4 y(1)=6 D)magnitude of error with 2 steps = 1 magnitude of error with 4 steps = 0.5