12.5E: 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 \[y'+p(x)y = q(x), \quad y(x_0)=y_0 \nonumber \] 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 \((a,b)\).