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

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## 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 $y'+p(x)y = q(x), \quad y(x_0)=y_0 \nonumber$ 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 $y = \left\{ \begin{array}{cl} (x^2-1)^{5/3}, & -1 < x < 1, \\[4pt] 0, & |x| \ge 1, \end{array} \right. \nonumber$ is a solution of the initial value problem $y'={10\over 3}xy^{2/5}, \quad y(0)=-1 \nonumber$ on $$(-\infty,\infty)$$. HINT: You'll need the definition $y'(\overline{x})=\lim_{x\to\overline{x}}\frac{y(x)-y(\overline{x})}{x-\overline{x}} \nonumber$ 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 $y = \left\{ \begin{array}{cl} \epsilon_1(x^2-a^2)^{5/3}, & - \infty < x < -a, \\[4pt] 0, & -a \le x \le -1, \\[4pt] (x^2-1)^{5/3}, & -1 < x < 1, \\[4pt] 0, & 1 \le x \le b, \\[4pt] \epsilon_2(x^2-b^2)^{5/3}, & b < x < \infty, \end{array} \right. \nonumber$ 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 $y'=y^{2/5}, \quad y(0)=1 \nonumber$ 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 $y'=3x(y-1)^{1/3}, \quad y(0)=1 \nonumber$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 $y'=3x(y-1)^{1/3}, \quad y(0)=9 \nonumber$ 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} \nonumber 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 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.