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Mathematics LibreTexts

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

  • Page ID
    18248
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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 \((a,b)\).