Answer: Use the Nonlinear Shooting method with TOL = 104 to approximate the solution to

Chapter 11, Problem 4

(choose chapter or problem)

Use the Nonlinear Shooting method with \(T O L=10^{-4}\) to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results.

a. \(y^{\prime \prime}=y^{3}-y y^{\prime}, \quad 1 \leq x \leq 2, y(1)=\frac{1}{2}, y(2)=\frac{1}{3}\); use h = 0.1; actual solution \(y(x)=(x+1)^{-1}\).

b. \(y^{\prime \prime}=2 y^{3}-6 y-2 x^{3}, \quad 1 \leq x \leq 2, y(1)=2, y(2)=\frac{5}{2}\); use h = 0.1; actual solution \(y(x)=x+x^{-1}\).

c. \(y^{\prime \prime}=y^{\prime}+2(y-\ln x)^{3}-x^{-1}, \quad 2 \leq x \leq 3, y(2)=\frac{1}{2}+\ln 2, y(3)=\frac{1}{3}+\ln 3\); use h = 0.1; actual solution \(y(x)=x^{-1}+\ln x .\)

d. \(y^{\prime \prime}=2\left(y^{\prime}\right)^{2} x^{-3}-9 y^{2} x^{-5}+4 x, \quad 1 \leq x \leq 2, y(1)=0, y(2)=\ln \ 256\); use h = 0.05; actual solution \(y(x)=x^{3} \ln x\).