In: Advanced Math
1.
Use Euler's method with step size 0.50.5 to compute the approximate yy-values y1≈y(0.5), y2≈y(1),y3≈y(1.5), and y4≈y(2) of the solution of the initial-value problem
y′=1+3x−2y, y(0)=2.
y1=
y2=
y3=
y4=
2.
Consider the differential equation dy/dx=6x, with initial condition y(0)=3
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)≈
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?)