Page 101 - Schaum's Outline of Differential Equations

84 SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS [CHAR 9

Characteristic equations for differential equations having dependent variables other than y are obtained
analogously, by replacing thej'th derivative of the dependent variable by A/ (j = 0, 1, 2).
The characteristic equation can be factored into

THE GENERAL SOLUTION
The general solution of (9.1) is obtained directly from the roots of (9.3). There are three cases to consider.
Case 1. AI and A^ both real and distinct. Two linearly independent solutions are e~* and e^, and
l
the general solution is (Theorem 8.2)

In the special case A 2 = -A x, the solution (9.4) can be rewritten as y = k^ cosh A,^ + k 2 sinh A,^.
Case 2. A^ = a + ib, a complex number. Since a^ and a 0 in (9.1) and (9.2) are assumed real, the roots
of (9.2) must appear in conjugate pairs; thus, the other root is A 2 = a — ib. Two linearly independent
(a
solutions are e^ a + lb)x and e ~ lb)x , and the general complex solution is

which is algebraically equivalent to (see Problem 9.16)

Case 3. A^ = A^. Two linearly independent solutions are e^ x and xe '*, and the general solution is

Warning: The above solutions are not valid if the differential equation is not linear or does not have constant
2
coefficients. Consider, for example, the equation y" — x y = 0. The roots of the characteristic equation are 'k l=x
and A^ = -x, but the solution is not

Linear equations with variable coefficients are considered in Chapters 27, 28 and 29.

Solved Problems

9.1. Solve
2
The characteristic equation is X - X- 2 = 0, which can be factored into (A,+ 1)(A, — 2) = 0. Since the roots
A,j = -1 and X 2 = 2 are real and distinct, the solution is given by (9.4) as

9.2. Solve
2
The characteristic equation is X - 7k = 0, which can be factored into (k - 0)(X - 7) = 0. Since the roots A,j = 0
and X 2 = 7 are real and distinct, the solution is given by (9.4) as