Page 169 - Schaum's Outline of Differential Equations

152 REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS [CHAP. 17

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

Furthermore, if we also define

then the initial conditions take the form x(t 0) = c, where = n.
t 0
17.4. Convert the differential equation X - 6i + 9x = t into the matrix equation

Here we omit Step 5, because the differential equation has no prescribed initial conditions. Following Step 1,
we obtain

Hence a^t) = 6, a Q(t) = —9, and/(f) = t. If we define two new variables, x^t) = x and x 2(t) = x, we have

Thus,

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

17.5. Convert the differential equation

into the matrix equation x(?) = A(?)x(t) + f(t).
The given differential equation has no prescribed initial conditions, so Step 5 is omitted. Following Step 1, we
obtain

Defining Xi(t) = x, x 2(t) = x, and x 3(t) = x (the differential equation is third-order, so we need three new variables),
we have that Xi = x 2, and x 2 = x 3, Following Step 3, we find

Thus,