Page 102 - Schaum's Outline of Differential Equations

CHAP. 9] SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS 85

9.3. Solve /' -5y = 0.
The characteristic equation is , which can be factored into Since the root!
and are real and distinct, the solution is given by (9.4) as

9.4. Rewrite the solution of Problem 9.3 in terms of hyperbolic functions.
Using the results of Problem 9.3 with the identities
and
we obtain,

where

9.5. Solve y + 10y + 21y = 0.
Here the independent variable is t. The characteristic equation is

which can be factored into

The roots A,j = -3 and X 2 = -7 are real and distinct, so the general solution is

9.6. Solve x - O.Olx = 0.
The characteristic equation is

which can be factored into

The roots A,j = 0.1 and X 2 = -0.1 are real and distinct, so the general solution is

or, equivalently,

9.7. Solve /' + 4/ + 5y = 0.
The characteristic equation is

Using the quadratic formula, we find its roots to be

These roots are a complex conjugate pair, so the general solution is given by (9.6) (with a = -2 and b = 1) as