LINEAR DIFFERENTIAL EQUATIONS
CHAP. 3]
89
which is equivalent to the solution obtained in Problem 3.36. Note that for this equation it is not necessary to consider Case 2, p = 1/2 since we have already obtained the required solution. Note also that if we had considered ft — 1/2 first, we would not have obtained the general solution. This is typical in general when the roots of the indicial equation differ by any integer except zero. In other equations both cases would have to be considered, each leading to a series solution. The general solution would then be obtained from these series on multiplying each by an arbitrary constant and adding.
SIMULTANEOUS EQUATIONS 3.41. Solve the system of equations
Write the system as (.D2 + 3)« + Dy = e~t,
-Wx + (D2 + B)y = sin2t
(1)
Method 1, using determinants.
Formal application of Cramer's rule for solving linear equations can be used and we find
where we have expanded the determinants so that operators precede functions. Note that the results are equivalent to (Z>2 + i)(D2 + g)x = 4e-«-2cos2t, (D2 + 1)(Z>2 + 9)j/ = -sin2i-4e-' Solving these equations we find
We can show that the total number of arbitrary constants in the solution is the same as the degree of the polynomial in D obtained from the determinant #2 + 3 D -W Z>2 + 3
= D* + 1002 + 9
i.e. 4. Thus there is a relationship between the constants Cj, e2, cs, c4 and cs, ce, c7, cg. To determine this relationship we must substitute the values of x and y obtained above in the original equations. If we do, we find c5 = 2c2, ce = -2c1( c7 = -2c4, c8 = 2c8 Thus the required solution is
Method 2.
We can also eliminate one of the variables, for example x, by operating on the first of equations (Í) with 4Z>, the second with D2 + 3 and adding. To solve the resulting equations, we can then use the method of undetermined coefficients.