1550078481-Ordinary_Differential_Equations__Roberts_

324 Ordinary Differential Equations

If the initial conditions were specified at c = 2, then the interval of exis-

tence and uniqueness would be (7r /2, 37r /2), since this is the largest interval

containing c = 2 on which all the functions aij ( x) and bi ( x) are simultane-

ously defined and continuous. If the initial conditions were specified at c = .5,

then the interval of existence and uniqueness would be (0, 1). If the initial

condition were specified at any c < 0, then there would be no solution because

the system (15a) would be undefined at c, since a 1 2(x) = ..jX is undefined for

x < o.

At this point the only question remaining to be answered from the coll ec-

tion of questions at the beginning of this section is "How can an n-th order

differential equation be rewritten as an equivalent system of first-order dif-

ferential equations?" Perhaps, we should first answer the ultimate question:

"Why is there a need to write an n-th order differential equation as a system

of n first-order differential equations?" The simple reason is because most

computer programs are written to so lve the general system initial value prob-

lem (11) and not an n-th order differential equation. The general n-th order

differential equation has the form

(16) y(n) = g(x, y, y(l), ... , y<n-1)).

Letting u 1 = y, u2 = y(ll, ... , Un = y<n-l), differentiating each of these

equations, and substituting for y, y(l), ... , y<n-l) in terms of u 1 , u 2 , ... , Un,

we see that equation (16) may be rewritten as the equivalent system

u~ U2 fi (x, U1, u2, ... , Un)

u' 2 U3 h (x, U1, U2, ... ) Un)

(17)

u~-1 = Un fn-1(x, u1, u2, ... , Un)

u' n g(x, u1, u2, ... , Un) fn(X, U1, U2, ... , Un)·

Observe that this system is a special case of the system (lla). The initial

conditions corresponding to (llb) are u 1 (c) = d 1 , u 2 (c) = d 2 , ... , and un(c) =

dn. When we apply the inverse transformation, these conditions in terms of

y and its derivatives become y(c) = d 1 , y(ll(c) = d 2 , y<^2 l(c) = d 3 , ... , and

y<n-ll(c) = dn. Hence, the general n-th order initial value problem

(18a) Y(n) = g(x, y, Y(l), ... , Y(n- 1))

(18b)

is equivalent to the system initial value problem