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Chapter 3: Q 3.7-10E (page 139)

Use the fourth-order Runge鈥揔utta algorithm to approximate the solution to the initial value problemy'=1-xy,y(1)=1at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.

Short Answer

Expert verified

蠒(2)=0.7014

Step by step solution

01

Find the values of ki, i = 1, 2, 3, 4

Since y'=1-xy,y(1)=1and x=x0=1,y=yo=1and h = 1, M = 10

k1=hf(x,y)=1(1-0)=0k2=hfx+h2,y+k12=-12k3=hfx+h2,y+k22=-18k4=hfx+h,y+k3=-34

02

Find the values of x and y

x=1+1=2y=1+16k1+2k2+2k3+k4=1+16(0-212-218-34)=0.6667

Therefore 蠒2=0.6667.

Since1-0.6667=0.3333>0.001

03

Find the other values

Apply the same procedure for h=0.5, and h=0.25 respectively.

蠒2=0.700799

Thus, 0.6667-0.7008=0.0341>0.001

蠒2=0.701385

Hence, 0.6667-0.7014=0.0006<0.001

So,蠒2=0.7014.

Hence the solution is 蠒(2)=0.7014

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