# 2.3.1: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)


## Q2.3.1

In Exercises 2.3.1-2.3.13, find all $$(x_0,y_0)$$ for which Theorem 2.3.1 implies that the initial value problem $$y'=f(x,y),\ y(x_0)=y_0$$ has (a) a solution and (b) a unique solution on some open interval that contains $$x_0$$.

1. $${y'={x^2+y^2 \over \sin x}}$$

2. $${y'={e^x+y \over x^2+y^2}}$$

3. $$y'= \tan xy$$

4. $${y'={x^2+y^2 \over \ln xy}}$$

5. $$y'= (x^2+y^2)y^{1/3}$$

6. $$y'=2xy$$

7. $${y'=\ln(1+x^2+y^2)}$$

8. $${y'={2x+3y \over x-4y}}$$

9. $${y'=(x^2+y^2)^{1/2}}$$

10. $$y' = x(y^2-1)^{2/3}$$

11. $$y'=(x^2+y^2)^2$$

12. $$y'=(x+y)^{1/2}$$

13. $${y'={\tan y \over x-1}}$$

## Q2.3.2

14. Apply Theorem 2.3.1 to the initial value problem $y'+p(x)y = q(x), \quad y(x_0)=y_0$ 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 $y = \left\{ \begin{array}{cl} (x^2-1)^{5/3}, & -1 < x < 1, \\[6pt] 0, & |x| \ge 1, \end{array} \right.$ is a solution of the initial value problem $y'={10\over 3}xy^{2/5}, \quad y(0)=-1$ on $$(-\infty,\infty)$$. HINT: You'll need the definition $y'(\overline{x})=\lim_{x\to\overline{x}}\frac{y(x)-y(\overline{x})}{x-\overline{x}}$ to verify that $$y$$ satisfies the differential equation at $$\overline{x}=\pm 1$$.
2. Verify that if $$\epsilon_i=0$$ or $$1$$ for $$i=1$$, $$2$$ and $$a$$, $$b>1$$, then the function $y = \left\{ \begin{array}{cl} \epsilon_1(x^2-a^2)^{5/3}, & - \infty < x < -a, \\[6pt] 0, & -a \le x \le -1, \\[6pt] (x^2-1)^{5/3}, & -1 < x < 1, \\[6pt] 0, & 1 \le x \le b, \\[6pt] \epsilon_2(x^2-b^2)^{5/3}, & b < x < \infty, \end{array} \right.$ is a solution of the initial value problem of a on $$(-\infty,\infty)$$.

16. Use the ideas developed in Exercise 2.3.15 to find infinitely many solutions of the initial value problem $y'=y^{2/5}, \quad y(0)=1$ on $$(-\infty,\infty)$$.

17. Consider the initial value problem $y' = 3x(y-1)^{1/3}, \quad y(x_0) = y_0. \tag{A}$

1. For what points $$(x_0,y_0)$$ does Theorem 2.3.1 imply that (A) has a solution?
2. For what points $$(x_0,y_0)$$ does Theorem 2.3.1 imply that (A) has a unique solution on some open interval that contains $$x_0$$?

18. Find nine solutions of the initial value problem $y'=3x(y-1)^{1/3}, \quad y(0)=1$that are all defined on $$(-\infty,\infty)$$ and differ from each other for values of $$x$$ in every open interval that contains $$x_0=0$$.

19. From Theorem 2.3.1, the initial value problem $y'=3x(y-1)^{1/3}, \quad y(0)=9$ has a unique solution on an open interval that contains $$x_0=0$$. 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 $y'=3x(y-1)^{1/3}, \quad y(3)=-7 \tag{A}$ has a unique solution on some open interval that contains $$x_0=3$$. Determine the largest such open interval, and find the solution on this interval.
2. Find infinitely many solutions of (A), all defined on $$(-\infty,\infty)$$.

21. Prove:

1. If $f(x,y_0) = 0,\quad a<x<b, \tag{A}$ and$$x_{0}$$ is in $$(a,b)$$, then $$y≡y_{0}$$ is a solution of \begin{aligned} y'=f(x,y),\quad y(x_{0})=y_{0}\end{aligned} on $$(a,b)$$.
2. If $$f$$ and $$f_y$$ are continuous on an open rectangle that contains $$(x_0,y_0)$$ and (A) holds, no solution of $$y'=f(x,y)$$ other than $$y\equiv y_0$$ can equal $$y_0$$ at any point in $$(a,b)$$.

This page titled 2.3.1: 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.