12.5E: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)
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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+3yx−4y
9. y′=(x2+y2)1/2
10. y′=x(y2−1)2/3
11. y′=(x2+y2)2
12. y′=(x+y)1/2
13. y′=tanyx−1
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.
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(y−1)1/3,y(x0)=y0.
18. Find nine solutions of the initial value problem y′=3x(y−1)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(y−1)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.
21. Prove: