Chapter 1.2, Problem 51E

The functions $y(x)=\frac{1}{16}{x}^4$, −∞ < x < ∞ and

$y(x)=\{\begin{array}{ll}0,\hfill & x<0\hfill \\ \frac{1}{16}{x}^4,\hfill & x\ge 0\hfill \end{array}$

have the same domain but are clearly different. See Figures 1.2.12(a) and 1.2.12(b), respectively. Show that both functions are solutions of the initial-value problem dy/dx = xy^1/2, y(2) = 1 on the interval (−∞, ∞). Resolve the apparent contradiction between this fact and the last sentence in Example 5.

FIGURE 1.2.12 Two solutions of the IVP in Problem 51