MTH203: Assignment-9

1. Find the eigen values and eigen functions of the following Strum-Liouville problems:

(i) y′′ + λy = 0, y(0) = y′(1) + y(1) = 0

(ii)♣ (xy′)′ + λx−1y = 0, y(1) = y′(e) = 0.

2. ♣ If p(x), q(x), r(x) are all greater than zero on (a, b), then prove that the eigen values of

the Strum-Liouville problem, (p(x)y′)′− q(x)y+λr(x)y = 0, are positive with any of the

boundary conditions: (i) p(a) = p(b) = 0, (ii) y(a)−ky′(a) = y(b)+my′(b) = 0, k,m > 0,

(iii) p(a) = p(b) with y(b) = y(a), y′(b) = y′(a).

3. ♣ Consider the Strum-Liouville problem

(p(x)y′)′ + [q(x) + λr(x)]y = 0

with p(x) > 0 on [a, b] and y(a) 6= y(b), y′(a) 6= y′(b). Show that every eigen function is

unique except for a constant factor.

4. If f is a piecewise continuous periodic function of period T , then show that∫ T

0f(x) dx =

∫ a+T

af(x) dx, (a is a constant).

5. Find the Fourier series of f (assuming f to be periodic with period 2π):

(a) f(x) =

{1 if 0 < x < π

−1 if π < x < 2π(b)♣ f(x) =

{x if −π/2 < x < π/2

0 if π/2 < x < 3π/2

(c) f(x) = x2/4, −π < x < π (d) f(x) = x, 0 < x < 2π

In each case, find the sum of the Fourier series at x = 101π/2.

6. Show that

(i) 1− 1/3 + 1/5− 1/7 + · · · · · · = π/4 [use 6(a)]

(ii) 1− 1/4 + 1/9− 1/16 + · · · · · · = π2/12 [use 6(c)]

7. ♣ Expand f(x) in a Fourier series on the interval [−2, 2] if f(x) = 0 for −2 ≤ x < 0 and

f(x) = 1 for 0 ≤ x ≤ 2. (Assume f to be periodic with period p = 2L = 4).

8. Find the Fourier cosine series as well as sine series for f(t) = 1 + sinπx, 0 < x < 1.

9. Using the Fourier integral representation, show that

(a)♣∫ ∞

0

cosxω + ω sinxω

1 + ω2dω =

0, x < 0,

π/2, x = 0,

πe−x, x > 0.

(b)

∫ ∞

0

cosxω

1 + ω2dω =

π

2e−x, x > 0

10. ♣ Find A(ω) such that1

π

∫ ∞

0

A(ω) cosxω dω =1

1 + x2.