1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 51

boundary of the rectangle twice instead of two different boundaries. T he

direction field for the differential equation y' = -x/y and the solution of the

IVP (10) are displayed in Figure 2.9.

2 /////______ ,__ ____________ """'

/////_______ -----------"""'"

/ / / / / /----------------"-.. "-..-....._,""

//////---/ / / / / .....-::~--------,..._---~---"""'"""' """"::--.....-....._,"" '\ \

I////./"/__..,... ________ "-'..::::~'\\\\

I I I / ~ / //-------........._ "-°" ~ \ \ \ \

II/Iii///__..,----"-"-'\~\\

I I I I I I / / --"-. \ \ \ \ \ \ \

y(x) 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ 2; ~ J ~ } } } ~ t

\ \ \ \ \ \ "--"----~---/ / / I I I I I

\\\\'\""-"----~---////////

• 1 \ \ \ '\ \ '\ "-....., '\ °""-"-"-"-.. ......... ____----- ---//////I // / / / / /
  '\ ""-....., "-.. ......... ___ /// / / /
  " "-....., "-.. "-.. -------------------/ / / / /
  "-....., "-.. "-.. -----------------------/ / / /


• 2 -.....,-....., "-.. "-.. ---------- ,___---------/ / / /
  
  
  • 2 -1 0 2
    x
  
  
  
  

Figure 2.9 Direction Field for the DE y' = -x/y and the

Solution of the IVP y' = -x/y; y(-l) = 1

When the function f(x, y) in the initial value problem (1) y' = f(x, y);

y(c) = d has the form f(x, y) = a(x)y + b(x), the initial value problem is

call ed a linear first-order initia l value proble m; otherwise, the init ial

value problem is said to be nonlinear. The initial value problems (2), (5), and

(9) which we examined in examples 1, 2, and 3, respectively, are all nonlinear

first-order initial value problems. The intervals on which the solutions to these

problems existed and were unique exhibited no discernable pattern, mainly

because the initial value problems were nonlinear. However, when we apply

the fundamental existence and uniqueness theorem to the linear first-order

initial value problem

(11) y' = a(x)y + b(x) = f(x, y); y(c) = d

we obtain an existence and uniqueness theorem in which the interval of ex-
istence and uniqueness is completely determined by the functions a(x) and

b(x). Differentiating f(x, y) = a(x)y+b(x) partially with respect toy, we find

fy(x, y) = a(x). Hence, when the functions a(x) and b(x) are both defined

and continuous functions of x on the interval (a , (3), the functions f(x, y) and
fy(x, y) are both defined and continuous functions of x and y in any finite
rectangle R = {(x, y)I a< x < (3 and / < y < b}. So the linear IVP ( 11 )
has a unique solution and this solution can be extended uniquely until the
boundary of the largest rectangle on which both a(x) and b(x) are defined