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12.5E: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)

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Q2.3.1

In Exercises 2.3.1-2.3.13, find all (x0,y0) for which Theorem 2.3.1 implies that the initial value problem y=f(x,y), y(x0)=y0 has (a) a solution and (b) a unique solution on some open interval that contains x0.

1. y=x2+y2sinx

2. y=ex+yx2+y2

3. y=tanxy

4. y=x2+y2lnxy

5. y=(x2+y2)y1/3

6. y=2xy

7. y=ln(1+x2+y2)

8. y=2x+3yx4y

9. y=(x2+y2)1/2

10. y=x(y21)2/3

11. y=(x2+y2)2

12. y=(x+y)1/2

13. y=tanyx1

Q2.3.2

14. Apply Theorem 2.3.1 to the initial value problem y+p(x)y=q(x),y(x0)=y0 for a linear equation, and compare the conclusions that can be drawn from it to those that follow from Theorem 2.1.2.

15.

  1. Verify that the function y={(x21)5/3,1<x<1,0,|x|1, is a solution of the initial value problem y=103xy2/5,y(0)=1 on (,). HINT: You'll need the definition y(¯x)=limx¯xy(x)y(¯x)x¯x to verify that y satisfies the differential equation at ¯x=±1.
  2. Verify that if ϵi=0 or 1 for i=1, 2 and a, b>1, then the function y={ϵ1(x2a2)5/3,<x<a,0,ax1,(x21)5/3,1<x<1,0,1xb,ϵ2(x2b2)5/3,b<x<, is a solution of the initial value problem of a on (,).

16. Use the ideas developed in Exercise 2.3.15 to find infinitely many solutions of the initial value problem y=y2/5,y(0)=1 on (,).

17. Consider the initial value problem y=3x(y1)1/3,y(x0)=y0.

  1. For what points (x0,y0) does Theorem 2.3.1 imply that (A) has a solution?
  2. For what points (x0,y0) does Theorem 2.3.1 imply that (A) has a unique solution on some open interval that contains x0?

18. Find nine solutions of the initial value problem y=3x(y1)1/3,y(0)=1that are all defined on (,) and differ from each other for values of x in every open interval that contains x0=0.

19. From Theorem 2.3.1, the initial value problem y=3x(y1)1/3,y(0)=9 has a unique solution on an open interval that contains x0=0. Find the solution and determine the largest open interval on which it is unique.

20.

  1. From Theorem 2.3.1, the initial value problem y=3x(y1)1/3,y(3)=7 has a unique solution on some open interval that contains x0=3. Determine the largest such open interval, and find the solution on this interval.
  2. Find infinitely many solutions of (A), all defined on (,).

21. Prove:

  1. If f(x,y0)=0,a<x<b, andx0 is in (a,b), then yy0 is a solution of y=f(x,y),y(x0)=y0 on (a,b).
  2. If f and fy are continuous on an open rectangle that contains (x0,y0) and (A) holds, no solution of y=f(x,y) other than yy0 can equal y0 at any point in (a,b).

This page titled 12.5E: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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