DEPARTMENT OF MATHEMATICS & STATISTIS, IIT KANPUR
MTH 203, SEM. I, 2005-06
Assignment No.7
1. For each of the following differential equations, locate and classify its singular points on the axis:
a. b.
2. For each of the following equations, verify that the origin is a regular singular point and obtain two linearly independent solutions:
a. b.
c. d.
3. Show that the equation has only one Frobenius series solution. Find the general solution. (Note that it is Bessel’s eqn. of order 0).
4. Solve the I. V. P. .
5. By use of recurrence relations, show that (i)
(ii) , (iii)
6. Show that
(i)
(ii)
7. Express the following
(i) in terms of and (ii) in terms of and
(iii) in terms of and .
8. (a) Show that between any consecutive positive zeros of there is precisely one zero of and vice versa.
(b) Show that between any consecutive zeros of there is precisely one zero of and vice versa.
10. Let Y(x) and U(x) be solutions of equations (1) and (2) of problem no. 6, then
(a) Show that Y(x) and U(x) have zeros at the same points.
(b) Show that every nontrivial solution of has infinitely many zeros. Hence infer that, () has infinitely zeros.
11.* Let U(x) be a nontrivial solution of on [a, b], then show that U(x) has at most finitely many zeros on [a, b], and has at most one zero if on [a, b].
12.* Show that if , then every interval of length contains at most one zero of .
13. Problem # 4, 5, 7, 8 of Sec. 4.5 and # 4, 9, 10 of Sec. 4.6 from ‘Advanced Engineering Mathematics’ – E. Kreyszig.
14. Consider the differential equation .
a. Show that is an irregular singular point.
b. Use the fact that is a solution to find a second independent solution by the method of variation of parameters and show that the second solution cannot be expressed as a Frobenius series.
15. The differential equation has as an irregular singular point. If (3) is inserted in this equation, show that and the corresponding Frobenius series ‘solution’ is the power series ,
which converges only at This demonstrates that even when a Frobenius series formally satisfies such an equation, it is not necessarily a valid solution.