<exercise>
<statement>
<p>
For each of the following Initial Value Problems (IVPs), designate the following:
</p>
<ul>
<li>its Ordinary Differential Equation (ODE)</li>
<li>its Initial Value or Values (IVs)</li>
<li>the order of the IVP</li>
<li>its independent variable</li>
<li>its dependent variable</li>
<li>whether its solution is implicit or explicit</li>
</ul>
<p>
Then show how to verify that its solution is valid.
</p>
<ol>
<xsl:for-each select="ivps/*">
<li>
<ul>
<li>IVP:
<m>
<xsl:value-of select="ode"/>\hspace{2em}
<xsl:value-of select="y"/>(<xsl:value-of select="t0"/>)=
<xsl:value-of select="y0"/><xsl:if test="yp0">,
<xsl:value-of select="y"/>'(<xsl:value-of select="t0"/>)=
<xsl:value-of select="yp0"/></xsl:if>
</m>
</li>
<li>Solution: <m><xsl:value-of select="sol"/></m></li>
</ul>
</li>
</xsl:for-each>
</ol>
</statement>
<answer>
<ol>
<xsl:for-each select="ivps/*">
<li>
<ul>
<li>ODE: <m><xsl:value-of select="ode"/></m></li>
<li>IV(s): <m>
<xsl:value-of select="y"/>(<xsl:value-of select="t0"/>)=
<xsl:value-of select="y0"/><xsl:if test="yp0">,
<xsl:value-of select="y"/>'(<xsl:value-of select="t0"/>)=
<xsl:value-of select="yp0"/></xsl:if>
</m></li>
<li>Order: <xsl:value-of select="order"/></li>
<li>Independent variable: <m><xsl:value-of select="t"/></m></li>
<li>Dependent variable: <m><xsl:value-of select="y"/></m></li>
<li>The solution <m><xsl:value-of select="sol"/></m> is <xsl:value-of select="ex_im"/>.</li>
</ul>
</li>
</xsl:for-each>
</ol>
</answer>
</exercise>
