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Unformatted text preview: d a nonzero polynomial solution to this diﬀerential equation, in the case
where p = 3.
d. Find a basis for the space of solutions to the diﬀerential equation
(1 − x2 ) d2 y
dy
+ 12y = 0.
− 2x
2
dx
dx 1.2.3. The diﬀerential equation
(1 − x2 ) d2 y
dy
−x
+ p2 y = 0,
dx2
dx where p is a constant, is known as Chebyshev’s equation . It can be rewritten in
the form
d2 y
dy
+ Q(x)y = 0,
+ P (x)
2
dx
dx where
14 P (x) = − x
,
1 − x2 Q(x) = p2
.
1 − x2 a. If P (x) and Q(x) are represented as power series about x0 = 0, what is the
radius of convergence of these power series?
b. Assuming a power series centered at 0, ﬁnd the recursion formula for an+2
in terms of an .
c. Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤
9.
d. In the special case where p = 3, ﬁnd a nonzero polynomial solution to this
diﬀerential equation.
e. Find a basis for the space of solutions to
(1 − x2 ) d2 y
dy
+ 9y = 0.
−x
dx2
dx 1.2.4. The diﬀerential equation
− d2
+ x2 z = λz
dx2 (1.12) arises when treating the quantum mechanics of simple harmonic motion.
a. Show that making the substitution z = e−x
Hermite’s diﬀerential equation 2 /2 y transforms this equation into d2 y
dy
− 2x
+ (λ − 1)y = 0.
2
dx
dx
b. Show that if λ = 2n +1 where n is a nonnegative integer, (1.12) has a solution
2
of t...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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