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Diff Eq 2.4

 

<== Differential Equations
Number Question Answer
See the work
1. Use Euler's method with step size 0.5 to compute the approximate y-values y_1 \approx y(0.5), y_2 \approx y(1), y_3 \approx y(1.5), and y_4 \approx y(2) of the solution of the initial-value problem
y' = -2  - 4 x + 2 y, \ \ \ y(0)=4.

y1= ,
y2= ,
y3= ,
y4= .
y1= 7
y2= 12
y3= 21
y4= 38
The work

Point Cost: 3
2. Consider the differential equation
rac{dy}{dx} = 7 x,
with initial condition y(0) = 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 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 =

A)
2 step=
h=0.5
y(1)= 15/4

4 step=
h=0.25
y(1)=37/8


B)
y=7x+2


C)
2 step= 1.75
4 step= 0.875

D) 2

The work

Point Cost: 3