functions and vice versa-a result of the series alone. Finally, we emphasize that it is not particularly important if, as in Example are unable to determine the general coefficient a, in terms of ao and a1. What is essential is that we can determine as many coefficients as we want. Thus we can find as many terms in the two series solutions as we want, even if we cannot determine the general term. While the task of calculating several coefficients in a power series solution is not difficult, it can be tedious, A symbolic manipulation package can oe teiprur nere; some are able to find a specified number of terms in a power series otddon in response to a single command, Wwith a suitable graphics package one can also produce plots such as those shown in the figures in this section. In each of Problems 1 through 14 solve the given differential equation by means of a power series about the given point xo. Find the recurrence relation; also find the first four terms in cach of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution. PROBLEMS 1. y"-y = 0, 3. y" – xy' - y = 0, 5. (1– x)y" + y = 0, (2. y" – xy' - y = 0, 4. y" + k²x²y = 0, 6. (2+x²)y" – xy' + 4y = 0, Xo =0 %3D Xo = 0, k a constant %3D Xo =1 0% %3D Xo = 0 Z y" + xy' +2y = 0, 9. (1+x²)y" – 4xy' + 6y = 0, 11. (3 – x²)y" – 3xy' – y = 0, Xo = 0 8. xy" +y' + xy = 0, %3D %3D Xo = 1 10. (4 – x²)y" + 2y = 0, %3D Xo = 0 Xo = 0 %3D 12. (1 – x)y" + xy' - y = 0, 14. 2y" + (x +1)y' + 3y = 0, Xo =0 Xo = 0 %3D 13. 2y" + xy' +3y = 0, Xo =0 %3D Xo = 2 %3D In each of Problems 15 through 18: (a) Find the first five nonzero terms in the solution of the given initial value problem. (b) Plot the four-term and the five-term approximations to the solution on the same axes. (c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate. 15. y"– xy' –y = 0, 216. (2+x²)y" – xy' +4y = 0, У () %3 2, у (0)%3D13; %3D see Problem 2 y(0) = -1, y'(0) = 3; %3D %3D see Problem 6