91Ó°ÊÓ

Show that the region of absolute stability for the trapezoidal method is the set of all complex \(h \lambda\) with Real \((\lambda)<0\).

(a) Using the Runge-Kutta method (6.10.8), solve $$ y^{\prime}=-y+x^{.1}[1.1+x] \quad y(0)=0 $$ whose solution is \(Y(x)=x^{1.1}\). Solve the equation on \([0,5]\), printing the errors at \(x=1,2,3,4,5\). Use stepsizes \(h=.1, .05, .025, .0125\), \(.00625 .\) Calculate the errors by which the errors decrease when \(h\) is halved. How does this compare with the usual theoretical rate of convergence of \(O\left(h^{2}\right) ?\) Explain your results. (b) What difficuity arises when trying to use a Taylor method of order \(\geq 2\) to solve the equation of part (a)? What does it tell us about the solution?