91Ó°ÊÓ

In each of the following cases, show that the given function \(Y(x)\) satisfies the associated differential equation. Then determine the value of \(c\) required by the initial condition. Finally, with reference to the general format in (8.7), identify \(f(x, z)\) for each differential equation. (a) \(\quad Y^{\prime}(x)=-Y(x)+\sin (x)+\cos (x), \quad Y(0)=1 ; \quad Y(x)=\sin (x)+c e^{-x}\) (b) \(Y^{\prime}(x)=\left(Y(x)-Y(x)^{2}\right) / x, \quad Y(1)=2 ; \quad Y(x)=x /(x+c), \quad x>0\) (c) \(Y^{\prime}(x)=\cos ^{2}(Y(x)), \quad Y(0)=\pi / 4 ; \quad Y(x)=\tan ^{-1}(x+c)\) (d) \(\quad Y^{\prime}(x)=Y(x)[Y(x)-1], \quad Y(0)=\frac{1}{2} ; \quad Y(x)=1 /\left(1+c e^{x}\right)\)

Verify that any function of the form \(Y(x)=c_{1} x+c_{2} x^{2}\) satisfies the equation $$ x^{2} Y^{\prime \prime}(x)-2 x Y^{\prime}(x)+2 Y(x)=0 $$ Determine the solution of the equation with the boundary conditions $$ Y(1)=0, \quad Y(2)=1 $$ Use the MATLAB program ODEBVP to solve the boundary value problem for \(h=\) \(0.1,0.05,0.025\), print the errors of the numerical solutions at \(x=1.2,1.4,1.6\) and 1.8. Comment on how errors decrease when \(h\) is halved. Do the same for the extrapolated solutions.

(a) Using the Runge-Kutta method (8.81), solve. $$ Y^{\prime}(x)=-Y(x)+x^{0.1}[1.1+x], \quad Y(0)=0 $$ whose solution is \(Y(x)=x^{1.1} .\) Solve the equation on \([0,5]\), printing the solution and the errors at \(x=1,2,3,4,5\). Use stepsizes \(h=0.1,0.05,0.025\), \(0.0125,0.00625 .\) Calculate the ratios by which the errors decrease when \(h\) is halved. How does this compare with the theoretical rate of convergence of \(O\left(h^{2}\right) .\) Explain your results as best as you can. (b) What difficulty arises in attempting to use a Taylor method of order \(\geq 2\) to solve the equation of part (a)? What does it tell us about the solution?