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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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n —0
5.7 Series Solutions near a Regular Singular Point, Part II
Now let us consider the general problem of determining a solution of the equation
L [y] = x2 y" + x [xp(x )]y' + [x 2q (x )]y = 0, (1)
11Edmond Nicolas Laguerre (1834-1886), a French geometer and analyst, studied the polynomials named for him about 1879.
5.7 Series Solutions near a Regular Singular Point, Part II
273
where
õð(õ) = ^2 ÐïÕï, õ 2q (õ) = ^2 ×ïÕï, (2)
ïï ï=0 ï=0
and both series converge in an interval |x | < p for some p > 0. The point õ = 0 is a regular singular point, and the corresponding Euler equation is
õ 2 y" + Ð0 õó' + q0 y =0 (3)
We seek a solution of Eq. (1) for õ > 0 and assume that it has the form
CO CO
Ó = Ô(Ã, õ) = õ^ àïõï = Y, àïõÃ+ï, (4)
ï=0 ï=0
where à0 = 0, and we have written y = ô(ã, õ) to emphasize that ô depends on r as well as õ .It follows that
OO
Ó = ? (r + ï)àïõã +ï-1, y” = Y. (r + ï)(ã + ï - 1)àïõã +ï-2. (5)
ï=0 ï=0
Then, substituting from Eqs. (2), (4), and (5) in Eq. (1) gives
a0r(r - 1)õã + a1(r + 1)ãõã+1 + ••• + àï(r + ï)(ã + ï - 1)õã+ï + ¦¦¦
+ (Ð0 + Ð\õ + ¦¦¦ + Ðïõï + •'•)
õ [à^õ’’ + à1 (r + 1)õã+1 +-+ àï (r + ï)õã+ï +-]
+ (q0 + q1 õ + ¦¦¦ + qïõï + ¦¦¦)
õ (à0õã + à1 õã+1 +----+ àïõã+ï +-) = 0.
Multiplying the infinite series together and then collecting terms, we obtain à0 F (r )õã + [à1 F (r + 1) + à^Ð^ + q^^+1
+ {à2 F (r + 2) + ao(Ð2r + q2) + àl^l(r + 1) + ql]}õr+2 + ¦ ¦ ¦ + HF (r + ï) + à0(Ðïr + qï) + à1[Ðï-( + 1) + qï-l]
+ ¦ ¦ ¦ + aï-1\p1(r + ï — 1) + q1]}õr +ï + ¦ ¦ ¦ = °>
or in a more compact form,
L\Ô](ã, õ) = a0F(r)õã
O ï-1
+ Y, \F(r + ï)àï + Ë ak[(r + k¿Ðï-k + qï-k] ã õ’+ï =0, (6)
ï=1 Ó k=0 J
where
F(r) = r (r - 1) + ð0ã + q0. (7)
For Eq. (6) to be satisfied identically the coefficient of each power of õ must be zero.
Since a0 = 0, the term involving õã yields the equation F(r) = 0. This equation is called the indicial equation; note that it is exactly the equation we would obtain in looking for solutions y = õã of the Euler equation (3). Let us denote the roots of the indicial equation by r1 and r2 with r1 > r2 if the roots are real. If the roots are complex, the designation of the roots is immaterial. Only for these values of r can we expect to find solutions of Eq. (1) of the form (4). The roots r1 and r2 are called the
274
Chapter 5. Series Solutions of Second Order Linear Equations
exponents at the singularity; they determine the qualitative nature of the solution in the neighborhood of the singular point.
Setting the coefficient of xr+n in Eq. (6) equal to zero gives the recurrence relation
F(r + n)an + Y ak[(r + k)pn-k + ?n-k] = °> n — L (8)
k=0
Equation (8) shows that, in general, an depends on the value of r and all the preceding coefficients a°, ax,, an_j. It also shows that we can successively compute av a2,..., an,... in terms of a° and the coefficients in the series for xp(x) and x2q (x) provided that F(r + 1), F(r + 2),..., F(r + n),... are not zero. The only values of r for which F(r) = 0 are r = rx and r = r2; since rx it follows that rx + n is not equal to rx or r2 for n — 1. Consequently, F(rr + n) = 0 for n — 1. Hence we can always determine one solution of Eq. (1) in the form (4), namely,
yl(x) = xr1
1 +¨ an (ri)xn
n1
x > 0. (9)
Here we have introduced the notation an (rx) to indicate that an has been determined from Eq. (8) with r = rr To specify the arbitrary constant in the solution we have
taken a0 to be 1.
If r2 is not equal to rx, and rl — r2 is not a positive integer, then r2 + n is not equal to rx for any value of n — 1; hence F(r2 + n) = 0, and we can also obtain a second solution
y2(x) = xr2
1 + ? an (r2)x
n = 1
x > 0. (10)
Just as for the series solutions about ordinary points discussed in Section 5.3, the series in Eqs. (9) and (10) converge at least in the interval |x| < p where the series for both xp(x) and x2q(x) converge. Within their radii of convergence, the power series
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