Stuck on this problem, want explanation. Will rate for full explanation!!! THX C

Question

Stuck on this problem, want explanation. Will rate for full explanation!!! THX

Consider the differential equation dy/dx = 4x, with initial condition y(0) = 3. Use Euler's method with two steps to estimate y when x = 1: (Be sure not to round your calculations at each step!) Now use four steps: (Be sure not to round your calculations at each step!) What is the solution to this differential equation (with the given initial condition)? 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 =

Explanation / Answer

A) dy /dx = 4x

with 2 steps,

h = 1/2

hence

y(1/2) = 3 + 1/2 * 4*0 = 3

y(1) = 3 + 1/2 * 4 * 1/2 = 4

with 4 steps,

h = 1/4

y(1/4) = 3 + 1/4 * 4*0 = 3

y(1/2) = 3 + 1/4 * 4* 1/4 = 13/4

y(3/4) = 13/4 + 1/4*4*1/2 = 15/4

y(1) = 15/4 + 1/4 * 4*3/4 = 9/2 = 4.5

B)

dy /dx = 4x

dy = 4x dx

integrating,

y(x) - y(0) = 2 x^2

y(x) = 3 + 2 x^2

C)

error with 2 steps = 1

eroor with 4 steps = 0.5

D)

factor = 2

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