Question

Please answer all parts to the problem highlighted in yellow (# 3). Thanks! Please show all steps and I will give a positive rating!

tep size (h/2, say. thereby suggesting their ac of some hidden difficulty i accurate and powerful met chapter Problems 2.4 In Problems 1 through 10, an initial value problem and its ex act solution y (r) are given. Apply Euler's method twice to approximate to this solution on the interval [0. 1], first with step size h0.25, then with step size h 0.1. Compare the three-decimal-place values of the two approximations at x nner are ith the value y(5) of the actual solution. decided 2.4.11 ve that us that 2 eld for of the at this using oo as ones) uler's es the ote: The application following this problem set lists illus- trative calculator/computer programs that can be used in the remaining problems programmable calculator or a computer will be useful for roblems 11 through 16. In each problem find the exact so- uion of the given initial value problem. Then apply Euler's od twice to approximate (to four decimal places) this so- tion on the given interval, first with step size h -0.01, then 26. with step size h = 0.005. Make a table showing the approxi- hate values and the actual value, together with the percentage meth rror for in the more accurate approximation, for x an integral ultiple of 0.2. Throughout respect to x. e of 0.2. Throughout, primes denote derivatives with

Answer 1

y_exact=2exp(1/2)-1~2.297

h=0.25,y(0)=1

dy/dx=f(x,y)=y+1

y(0.25)=y(0)+0.25f(y(0),x(0))=1+0.25(1+1)=1.5

y(0.5)=y(0.25)+0.25f(y(0.25),x(0.25))=1.5+0.25(1.5+1))=2.125

h=0.1

y(0.1)=y(0)+0.1f(y(0),x(0))=1+0.1(1+1)=1.2

y(0.2)=y(0.1)+0.1(1.2+1)=1.2+0.22=1.42

y(0.3)=y(0.2)+0.1(1.42+1)=1.42+0.242=1.662

y(0.4)=y(0.3)+0.1(1.662+1)=1.662+0.2662=1.9282

y(0.5)=y(0.4)+0.1(1.9282+1)=1.9282+0.29282=2.22102

Approximating to three decimal places

y(0.5)=2.221 with h=0.1

y(0.5)=2.125 with h=0.25

2.297 being the exact value.

We see a finer grid with h=0.1 gives a value accurate to one decimal place