Q8. For the differential equation x2 + x - (secx) y = 0, (a) x=0 is a irregular singular (b)x=0 is a regular singular point. (c) x=0, (d)x=1is a regular singular point.
Answers
Answer:
Step-by-step explanation:
Hence, if we omit the constant multiplier a0
, one solution of Eq. (8) is
y1
(x) = x
1 +
∞
n=1
(−1)
n
x
n
[3 · 5 · 7 ···(2n + 1)]n!
, x > 0. (19)
To determine the radius of convergence of the series in Eq. (19) we use the ratio test:
lim
n→∞
an+1
x
n+1
an
x
n
= lim
n→∞
|x|
(2n + 3)(n + 1)
= 0
for all x. Thus the series converges for all x.
Corresponding to the second root r = r
2 =
1
2
, we proceed similarly. From Eq. (17)
we have
an = −
an−1
2n(n −
1
2
)
= −
an−1
n(2n − 1)
, n ≥ 1.
Hence
a1 = −
a0
1 · 1
,
a2 = −
a1
2 · 3
=
a0
(1 · 2)(1 · 3)
,
a3 = −
a2
3 · 5
= −
a0
(1 · 2 · 3)(1 · 3 · 5)
,
and in general
an =
(−1)
n
n![1 · 3 · 5 ···(2n − 1)]
a0
, n ≥ 1. (20)
Again omitting the constant multiplier a0
, we obtain the second solution
y2
(x) = x
1/2
1 +
∞
n=1
(−1)
n
x
n
n![1 · 3 · 5 ···(2n − 1)]
, x > 0. (21)
As before, we can show that the series in Eq. (21) converges for all x. Since the leading
terms in the series solutions y1
and y2
are x and x
1/2
, respectively, it follows that the
solutions are linearly independent. Hence the general solution of Eq. (8) is
y = c1
y1
(x) + c2
y2
(x), x > 0.
The preceding example illustrates that if x = 0 is a regular singul