Page 104 - Schaum's Outline of Differential Equations
P. 104
CHAP. 9] SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS 87
The roots are real and equal, so the general solution is given by (9.7) as
9.13. Solve y" = 0.
2
The characteristic equation is A = 0, which has roots A : = A 2 = 0. The solution is given by (9.7) as
9.14. Solve x + 4x + 4x = 0.
The characteristic equation is
which can be factored into
The roots A : = A^ = -2 are real and equal, so the general solution is
9.15. Solve
Dividing both sides of the differential equation by 100, to force the coefficient of the highest derivative to be
unity, we obtain
Its characteristic equation is
which can be factored into
The roots A : = A^ = 0.1 are real and equal, so the general solution is
9.16. Prove that (9.6) is algebraically equivalent to (9.5).
Using Euler's relations
we can rewrite (9.5) as
where c 1 = d 1 + d 2 and c 2 = i^ - d 2).
Equation (_/) is real if and only if Cj and c 2 are both real, which occurs, if and only if di and d 2 are com-
plex conjugates. Since we are interested in the general real solution to (9.1), we restrict di and d 2 to be a
conjugate pair.
