Austin Cholley Jones MAT 275 ONLINE B Spring 2021
Assignment Section 2.7 Euler Method due 03/17/2021 at 11:59pm MST
1. (1 point)
Use Euler’s method with step size 0.5 to compute the approx-
imate y-values y1≈y(0.5),y2≈y(1),y3≈y(1.5),and y4≈y(2)
of the solution of the initial-value problem
y0=−2+5x+2y,y(0) = 5.
y1=,
y2=,
y3=,
y4=.
Answer(s) submitted:
•
•
•
•
(incorrect)
Correct Answers:
•9
•18.25
•38
•78.75
2. (1 point)
Consider the differential equation
dy
dx =2x,
with initial condition y(0) = 3.
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?)
Solution:
SOLUTION
A. Euler’s method with two steps uses ∆x=0.5, so we have
y(0) = 3,
y(0.5)≈y(0) + 0.5(2(0)) = 3, and
y(1)≈y(0.5) + 0.5(2(0.5)) = 3+0.5=3.5.
Similarly, with four steps ∆x=0.25, so
y(0) = 3,
y(0.25)≈y(0) + 0.25(2(0)) = 3,
y(0.5)≈y(0.25) + 0.25(2(0.25)) = 3.125,
y(0.75)≈y(0.5) + 0.25(2(0.5)) = 3.375,
y(1)≈y(0.75) + 0.25(2(0.75)) = 3.75,
B. We can just integrate to find the solution to the differential
equation: y=1x2+C. For y(0) = 3, we must have y=1x2+3.
C. The error in the two Euler approximations is just the dif-
ference between the estimates for y(1)and the exact value. That
is, the difference between 3.5 and 4, and between 3.75 and 4.
Thus, taking the magnitude of the differences,
Error in Euler with two steps = 0.5, and
Error in Euler with four steps = 0.25.
D. Since we doubled the number of steps we took, we expect
the error to drop by a factor of two (that is, the error in the Euler
approximation with two steps should be twice as big as that with
four), which it is.
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
Correct Answers:
•3 + 0.25*2
•3 + 0.25*2*(0.25 + 0.5 + 0.75)
•3 + (2/2)*xˆ2
•0.5
•0.25
•2
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 ap-
proximations you found?
Magnitude of error in Euler with 2 steps =
Magnitude of error in Euler with 4 steps =
1
