Page 128 - Schaum's Outline of Differential Equations
P. 128
CHAP. 13] INITIAL-VALUE PROBLEMS 111
13.2. Solve
The general solution of the differential equation is given in Problem 12.3 as
Therefore,
Applying the first initial condition to (_/), we obtain
or (noting that In 1 = 0),
Applying the second initial condition to (2), we obtain
or
Solving (3) and (4) simultaneously, we find that c l = —c 3 = (e— l)le. Substituting these values into (_/), we obtain
the solution of the initial-value problem as
13.3. Solve
Here yh = e (cj cos 2x + c 2 sin 2x), and, by the method of undetermined coefficients,
Thus, the general solution to the differential equation is
Therefore,
Applying the first initial condition to (_/), we obtain
Applying the second initial condition to (2), we obtain
Solving (3) and (4) simultaneously, we find that c l = 69/65 and c 2 = 131/130. Substituting these values into (_/),
we obtain the solution of the initial-value problem as
