Page 284 - Schaum's Outline of Differential Equations

CHAP. 27] LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 267

Substituting these two results into (_/) and letting c 1 = a 0 and c 2 = a 1; we obtain, as before,

27.7. Determine whether x = 0 is an ordinary point of the differential equation

2
Dividing by 2x , we have

As neither function is analytic at x = 0 (both denominators are zero there), x = 0 is not an ordinary point but, rather,
a singular point.

27.8. Determine whether x = 0 is an ordinary point of the differential equation

2
Here P(x) = 2/x and Q(x) = IIx. Neither of these functions is analytic at x = 0, so x = 0 is not an ordinary point
but, rather, a singular point.

27.9. Find a recurrence formula for the power series solution around t = 0 for the differential equation

Both P(t) = t — 1 and Q(t) = 2t — 3 are polynomials; hence every point, in particular t = 0, is an ordinary point.
Substituting Eqs. (27.5) through (27.7) into the left side of the differential equation, with t replacing x, we have

or

Equating each coefficient to zero, we obtain

In general,

which is equivalent to

Equation (2) is the recurrence formula for this problem. Note, however, that it is not valid for n = 0, because a_i
is an undefined quantity. To obtain an equation for n = 0, we use the first equation in (_/), which gives
a ^ = \a^ +|a 0