Page 289 - Schaum's Outline of Differential Equations

272 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS [CHAP. 27

Substituting back t = x+ 1, we obtain as the solution to the original problem

x
27.20. Find the general solution near x= 1 of /' + (x - l)y = e .
We set t = x - 1, hence x = t + 1. As in Problem 27.18, , so the given differential equation may be
rewritten as

Its solution is (see Problems 27.16 and 27.17)

Substituting back t = x— 1, we obtain as the solution to the original problem

27.21. Solve the initial-value problem

Since the initial conditions are prescribed at x = 2, they are most easily satisfied if the solution to the differential
equation is obtained as a power series around this point. This has already been done in Eq. (_/) of Problem 27.18.
Applying the initial conditions directly to this solution, we find that a 0 = 5 and aj = 60. Thus, the solution is

27.22. Solve /' + xy' + (2x - l)y = 0; X~l) = 2, /(-I) = -2.
Since the initial conditions are prescribed at x = — 1, it is advantageous to obtain the general solution to the
differential equation near x = — 1. This has already been done in Eq. (_/) of Problem 27.19. Applying the initial
conditions, we find that a 0 = 2 and aj = - 2. Thus, the solution is