1550078481-Ordinary_Differential_Equations__Roberts_

54 Ordinary Differential Equations

In exercises 7-12 find all points ( c , d) where solutions to the initial

value problem consisting of the given differential equation and the

initial condition y(c) = d may not exist or may not be unique.

7. y' = x/y^2 8. y' = Vfj/x

9. y' = xy/(1- y)

11. y' = J(y - 4)/x

10. y' = (xy)I/3

12. y' = -y/x + y^114

In exercises 13-22 state the interval on which the solution to the

linear initial value problem exists and is unique.

13. 
    

    y' = 4y - 5; y(l) = 4

    
    


14. 
    

    y' + 3y = 1; y(-2) = 1

    
    


15. 
    

    y' = ay + b; y(c) = d where a, b, c, and dare real constants.

    
    


16. 
    

    y' = x^2 +ex - sinx; y(2) = -1

    
    

17.

18.

19.

20.

21.

22.

1

y' = xy+--· y(-5) = 0

1 + x^2 '

y

y' = - + cosx; y(-1) = 0

x

y

y' = - + tanx; y(n") = 0

x

I y ~

Y = -4 - x 2 +vx;

I y ~

y = -4 -x 2 +vx;

y' =(cot x)y + cscx;

y(3) = 4

y(l) = -3

y(n/2) = 1

x2 + n

23. Verify that Y1(x) = 1 and y2(x) = sin(-
    2

-) are both solut ions on

the interval (-ft, ft) of the init ia l value problem

y' = -xJ17; y(O) = 1.

Does this violate the fundamental existence and uniqueness theorem? Ex-

plain.

24. Verify that Y1 (x) = 9 - 3x and y 2 (x) = -x^2 /4 are both solutions of
    the initial value problem

y' = (-x+ Jx^2 +4y)/2; y(6) = -9.
Does this violate the fundamental existence and uniqueness theorem? Ex-
plain.