1550078481-Ordinary_Differential_Equations__Roberts_

330 Ordinary Differential Equations

5. Consider the system initial value problem

(25a)

2y1 Y2 1 1

• 
  
  
  
  • 
    
    
    
    • -3+-- -
      x x^2 x x^2
      Y2 I 2y1+1-6x
    
    
    
    
  
  
  
  

(25b) Y1(l) = -2, Y2(l) = -5.

a. Is the system of differential equations (25a) linear or nonlinear?

b. Apply the appropriate theorem from this chapter to determine the

interval on which a unique solution to the IVP (25) exists.

c. Verify that

(26) {y1(x) = -2x, y2(x) = -5x^2 +x -1}

is the solution to the initial value problem (25). What is the largest

interval on which the functions y 1 (x) and Y2(x) of (26) and their deriva-

tives are defined and continuous? Why is this interval not the same

interval as the one which the theorem guarantees existence and unique-

ness of the solution? Is the set {y 1 (x), y 2 (x)} a solution of the IVP (25)

on (-00,00)? Why or why not?

6. Consider the system initial value problem

I

Y1

5y1 + 4y2 _ 2 x

x x

-6y1 _ 5y2 + 5 x

x x

(27a)

I

Y2

(27b) Y1(-l) = 3, Y2(-l) = -3.

a. Is the system of differential equations (27a) linear or nonlinear?

b. Apply the appropriate theorem from this chapter to determine the

interval on which a unique solution to the IVP (27) exists.

c. Verify that

(28) {y1(x)=2x^2 +x--^2 , Y2(x)=-x^2 -x+-}^3

x x

is the solution to the initial value problem (27). On what intervals are

the functions Y1(x) and Y2(x) of (28) and their derivatives simultane-

ously defined and continuous? How do these intervals compare with

the interval that the appropriate theorem guarantees the existence of a

unique solution?