Problems In each of Problems 1 through 11: a. Seek power series solutions of the given differential equation about the given point xo; find the recurrence relation that the coefficients must satisfy. b. Find the first four nonzero terms in each of two solutions y₁ and y2 (unless the series terminates sooner). 1. 2. 3. 4. 5. 6. 7. 8. 9. c. By evaluating the Wronskian W[y1, y21(xo), show that yı and form a fundamental set of solutions. 2 d. If possible, find the general term in each solution. y" - y = 0, y" + 3y = 0, y" -xy' - y = 0, y" - xy' - y = 0, y" +k²x²y = 0, Xo = 0 xo = 0 xo = 0. xo = 1 xo = 0, k a constant xo = 0 Xo = 0 Xo = 1 (1-x)y"+y=0, y"+xy' + 2y = 0, xy" + y + xy = 0, (3-x²) y" - 3xy' - y = 0, xo = 0 10. 2y" + xy' + 3y = 0, xo = 0 11. 2y" + (x + 1) y' + 3y = 0, xo = 2 In each of Problems 12 through 14: a. Find the first five nonzero terms in the solution of the given initial-value problem. G 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. SPLIT 12. y" - xy' - y = 0, y(0) = 2, y'(0) = 1; see Problem 3 -13. y" + xy + 2y = 0, y(0) = 4, y'(0) = -1; see Problem 7 14. (1-x) y" + xy' - y = 0, y(0) = -3, y'(0) = 2 = t and assuming 15. a. By making the change of variable x - 1 o Taylor series in powers of t, find two series solutions 17. Show of Airy's eq of the text. 18. The H where is a important eq a. Finc about x solution. b. Obse or the o polynom 8, and 10 multiplic c. The E solution coefficier 19. Consider a. Show problem. b. Look a power s in x3 in th In each of Pro series solution thereby obtaini 5.2.4 (except th solution). G 20. y" + G 21. (4- G 22. !!