
# 2.3E: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)

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In Exercises 1-13, find all $$(x_0,y_0)$$ for which Theorem [thmtype: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 $$x_0$$.

[exer:2.3.1] $${y'={x^2+y^2 \over \sin x}}$$ &

[exer:2.3.2] $${y'={e^x+y \over x^2+y^2}}$$

[exer:2.3.3] $$y'= \tan xy$$ &

[exer:2.3.4] $${y'={x^2+y^2 \over \ln xy}}$$

[exer:2.3.5] $$y'= (x^2+y^2)y^{1/3}$$ &

[exer:2.3.6] $$y'=2xy$$

[exer:2.3.7] $${y'=\ln(1+x^2+y^2)}$$ &

[exer:2.3.8] $${y'={2x+3y \over x-4y}}$$

[exer:2.3.9] $${y'=(x^2+y^2)^{1/2}}$$ &

[exer:2.3.10] $$y' = x(y^2-1)^{2/3}$$

[exer:2.3.11] $$y'=(x^2+y^2)^2$$ &

[exer:2.3.12] $$y'=(x+y)^{1/2}$$

[exer:2.3.13] $${y'={\tan y \over x-1}}$$

[exer:2.3.14] Apply Theorem [thmtype: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.

[exer:2.3.15]

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 $$(-\infty,\infty)$$.
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 $$(-\infty,\infty)$$.

[exer:2.3.16] Use the ideas developed in Exercise [exer:2.3.15} to find infinitely many solutions of the initial value problem $y'=y^{2/5}, \quad y(0)=1$ on $$(-\infty,\infty)$$.

[exer:2.3.17] Consider the initial value problem $y' = 3x(y-1)^{1/3}, \quad y(x_0) = y_0. \eqno{\rm (A)}$

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

[exer:2.3.18] 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 $$x_0=0$$.

[exer:2.3.19] From Theorem [thmtype: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’s unique.

[exer:2.3.20]

1. From Theorem [thmtype:2.3.1}, the initial value problem $y'=3x(y-1)^{1/3}, \quad y(3)=-7 \eqno{\rm (A)}$ 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 $$(-\infty,\infty)$$.

[exer:2.3.21] Prove:

1. If $f(x,y_0) = 0,\quad a<x<b, \eqno{\rm (A)}$ and $$x_0$$ is in $$(a,b)$$, then $$y\equiv y_0$$ is a solution of $y'=f(x,y), \quad y(x_0)=y_0$ 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 $$(a,b)$$.