Consider the differential equation with initial condition y ( 0 ) = 4 . A. Use E
ID: 2961710 • Letter: C
Question
Consider the differential equation
with initial condition y(0)=4.
A. 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!)
B. What is the solution to this differential equation (with the given initial condition)?
y=
C. 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 =
D. 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?)
Only have question about D
Explanation / Answer
two steps mean dx = 0.5
so y(0.5) = y(0) + y'(0)*dx = 4 +0*0.5 = 4
so y(1) = y(0.5) + y'(0.5)*dx = 4 + 3*0.5*0.5 = 4 + 3/4 = 4.75
for steps means dx = 0.25
y(0.25) = y(0) + y'(0)*dx = 4 + 0*0.25 = 4
y(0.5) = 4 + 3*0.25*0.25 = 4.1875
y(0.75) = 4.1875 + 3*0.5*0.25=4.5625
y(1) = 4.5625 + 3*0.75*0.25= 5.125
b) y = 3x^2/2 + 4
c) error with 2 steps = (3/2 + 4) - 4.75= 0.75
with 4 steps = (3/2 + 4) - 5.125 = 0.375
d) since linear d.e. 4/2 = 2
0.75/0.375 = 2