1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 365

If y is an n x 1 column vector whose entries are functions of a single inde-

pendent variable x, then the derivative of y , denoted by y' or dy / dx, is the

n x 1 column vector whose entries are the derivatives of the corresponding

entries of y. That is, if

(

Y1(Y2(x) x) ) (y~(y~(x) x))

y(x) = : , then y'(x) = :.

Yn(x) y~(x)

Recall from chapter 7 that an initial value problem for a system of n first-

order differential equations is the problem of solving the system of n differen-

tial equations

(la)

Y~ = Ji ( x, Y1 , Y2 , · · · , Yn)

y; = f2(x, Y1, Y2, · · · , Yn)

Y~ = fn(x, y1,Y2,. · .,yn)

subject to the n constraints

(lb)

If we let

(

Ji (x, Y1, Y2, · · ·, Yn))

f2(x, Y1, Y2, · · ·, Yn)

f ( x, y) =. , and

fn(x, y1,y2,. · .,yn)

then using vector notation we can write the initial value problem (1) more
concisely as

(2) y' = f(x, y); y(c) = d.

Notice that the vector initial value problem (2) is very similar in appearance
to the scalar initial value problem

(3) y' = f(x, y); y(c) = d.

which we studied in chapters 2 and 3. In fact, when n = 1, the vector initial
value problem (2) is exactly the scalar initial value problem (3).

Also recall from chapter 7 that a linear system initial value problem is the
problem of solving the system of n linear first-order differential equations

y~ = a11(x)y1 + a12(x)y2 + · · · + a1n(x)yn + b1(x)