X3.ptx

<exercise>
  <statement>
    <p>
      Explain what the Existence and Uniqueness Theorem for First Order IVPs
      guarantees about the existence and uniqueness of solutions for the following
      IVP.
    </p>
    <me>
      y'=<xsl:value-of select="F"/>\hspace{2em}
      x(<xsl:value-of select="t0"/>)=<xsl:value-of select="y0"/>
    </me>
  </statement>
  <answer>
    <p><m>F(t,y)=<xsl:value-of select="F"/></m> is continuous at and nearby the
      initial value so a solution exists for a nearby interval.
      </p>
    <xsl:choose>
        <xsl:when test="unique">
          <p><m>F_y=<xsl:value-of select="Fy"/></m> is continous
              at and nearby the initial value so the solution is unique for
              a nearby interval.
            </p>
        </xsl:when>
        <xsl:otherwise>
          <p><m>F_y=<xsl:value-of select="Fy"/></m> is not continous
              (or even defined)
              at the initial value so the guaranteed solution may not be unique.
            </p>
        </xsl:otherwise>
      </xsl:choose>
  </answer>
</exercise>