Question

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value of y(2) where y is the solution to the initial value problem

y′=y−2x             y(0) = 1

Recall that Euler’s method says: Approximate values for the solution of the initial value problem

y′=F(x, y),y(x0) =y0 with step size h, at xn=xn−1+h, are

yn=yn−1+hF(xn−1, yn−1)

Fill in the table for steps of size h= 0.2.

n	xn	yn=yn-1+0.2F(xn-1,Yn-1	y'=F(xn,yn)
0	0	1
1	.2
2	.4
3
4
5
6
7
8
9
10

Graph the portion of the approximate solution curve you found above. It should look like a lot of line segments. The first segment has been given on the grid below:

(c) Suppose f(x) is an exact solution to the initial value problem above. Describe, with justification, the behavior off(x) as x→∞. Hint: Graphing a slope field may be helpful for this.

Answer 1

h =

   0.200000000000000

---------------------------------------------------------------------------------------------------------------
n       tn          y_Euler          dy/dx     
----------------------------------------------------------------------------------------------------------------
0        0.000000         1.000000         1.000000
1        0.200000         1.200000         0.800000
2        0.400000         1.360000         0.560000
3        0.600000         1.472000         0.272000
4        0.800000         1.526400        -0.073600
5        1.000000         1.511680        -0.488320
6        1.200000         1.414016        -0.985984
7        1.400000         1.216819        -1.583181
8        1.600000         0.900183        -2.299817
9        1.800000         0.440220        -3.159780
10        2.000000        -0.191736        -4.191736

Direction filreld plot

After solve ode we get solution y(x)=2(x+1)=e^x

so x tends to infinity y is infinity