Page 168 - Schaum's Outline of Differential Equations

CHAP. 17] REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS 151

Solved Problems

17.1. Put the initial-value problem

into the form of System (17.7).
Following Step 1, we write x = - 2x + 8x + e'; hence, a^t) = —2, a Q(t) = 8, and/(f) = e'. Then, defining x^t) = x
and x 2(t) = x (the differential equation is second-order, so we need two new variables), we obtain x l=x 2.
Following Step 3, we find

Thus,

These equations are equivalent to the matrix equation x(t) = A(t)x(t) + f(t) if we define

Furthermore, if we also define c = then the initial conditions can be given by x(t 0) = c, where = 0.
t 0
17.2. Put the initial-value problem

into the form of System (17.7).
Proceeding as in Problem 17.1, with ^ replaced by zero, we define

The differential equation is then equivalent to the matrix equation x(t) = A(t)x(t) + f(t), or simply x(t) = A(t)x(t),

since f(t) = 0. The initial conditions can be given by x(t Q) = c, if we define t Q = 1 and c =

17.3. Put the initial-value problem

into the form of System (17.7).
Following Step 1, we write x = —x + 3; hence, a^t) = 0, a Q(t) = -1, and/(f) = 3. Then defining x^t) = x and
x 2(t) = x, we obtain x l = x 2. Following Step 3, we find

Thus,