Legendre's differential equation is given - (1 − x²)y" − 2xy' + a(a + 1)y = 0, y = y(x), a ≤R, (1). Show that the general solution of (1) is y = a₁y₁ (x) + a₁y/₂(x), where and №₁(x) = 1- - a(a - 1) 2 -x² a(a + 1) Σ+1(1 2n n=2 №₂(x) = x + Σn=1 1 2n+1 _ a(a + 1¹)....(1 - (1 — ala+¹).….….. (1 · - 2 a(a + 1) (2n-2)(2n-1) a(a+1) 2n(2n-1) 2n+1 -)x² -) x ² n
Legendre's differential equation is given - (1 − x²)y" − 2xy' + a(a + 1)y = 0, y = y(x), a ≤R, (1). Show that the general solution of (1) is y = a₁y₁ (x) + a₁y/₂(x), where and №₁(x) = 1- - a(a - 1) 2 -x² a(a + 1) Σ+1(1 2n n=2 №₂(x) = x + Σn=1 1 2n+1 _ a(a + 1¹)....(1 - (1 — ala+¹).….….. (1 · - 2 a(a + 1) (2n-2)(2n-1) a(a+1) 2n(2n-1) 2n+1 -)x² -) x ² n
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:EXERCISE 1C
Legendre's differential equation is given
-
(1-x²)y" — 2xy' + a(a + 1)y = 0, y = y(x),a ≤R, (1).
Show that the general solution of (1) is
y = aoy₁ (x) + a₁₂3₂ (x),
where
and
y₁(x) = 1
a(a − 1)
2
+00
x² - Σ²
n2=2
a(a + 1)/
2n
a(a +1D).... 1.
6
Y₂(x) = x + Σ₁¹ (1 — ala+¹))... (1
+00
1
1
-
2n+1
2
a(a + 1)
(2n-2) (2n-1)'
a(a+1)
2n(2n-1)
-2n+1
--)x²
-)x²n
2n
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