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							<c val="#808000">NEW FEATURES</c>
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								<link href=".\..\..\toc_tuto.xmcd" title="Links to Tutorials" popup="false">tutorials</link>
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								<c val="#800000">New ODE Solvers and State Space Solver</c>
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				<phrase>new features</phrase>
				<phrase>features, in Mathcad</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Mathcad 14 features the following new and enhanced functions for solving ordinary differential equations (ODEs):</p>
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				<phrase>ODE solvers, new</phrase>
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					<link href="#new_solvers" title="Links to section" popup="false">New Solvers for Ordinary Differential Equations</link>
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					<link href="#odesolve" title="Links to section" popup="false">Changes to Odesolve</link>
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					<link href="#Radau" title="Links to section" popup="false">Enhancements to Radau</link>
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					<link href="#statespace" title="Links to section" popup="false">New State Space Solver</link>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">New Solvers for Ordinary Differential Equations</p>
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				<phrase>ODE solvers</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad 14 adds three new functions for solving ODEs:</p>
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					<b>
						<c val="#800000">Adams</c>
					</b>
					<b>,</b> based on Adams methods, is designed to solve non-stiff systems. The syntax for <b>Adams</b> is </p>
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					<inlineAttr font-weight="normal">Adams(init, t1, t2, npoints, D, [tol])</inlineAttr>
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				<phrase>Adams, ODE solver</phrase>
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					<b>
						<c val="#800000">BDF</c>
					</b>, which uses backward differentiation formulas, is designed to solve stiff systems. The syntax for <b>BDF</b> is</p>
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					<inlineAttr font-weight="normal">BDF(init, t1, t2, npoints, D, [J], [tol])</inlineAttr>
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				<phrase>BDF</phrase>
				<phrase>backward differentiation formulas</phrase>
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					<b>
						<c val="#800000">AdamsBDF</c>
					</b>, a hybrid solver that dynamically detects whether a system is non-stiff or stiff, and calls <b>Adams </b>or <b>BDF </b>accordingly. </p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The syntax for <b>AdamsBDF</b> is </p>
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					<inlineAttr font-weight="normal">AdamsBDF(init, t1, t2, npoints, D, [J], [tol])</inlineAttr>
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				<phrase>Adams/BDF</phrase>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Arguments</p>
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				<phrase>ODE solvers, arguments</phrase>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The required arguments, <b>init</b>, <b>t1, t2</b>, <b>npoints</b>, and <b>D</b>, for the new solvers are the same as for the existing solvers. See <b>
						<u>
							<link href=".\..\..\calculus_des/odeqs1.xmcd" title="Links to section" popup="false">Solving a First-Order ODE Initial Value Problem</link>
						</u>
					</b> for an example that uses the existing solver <b>rkfixed</b> with these arguments. In that example, you can replace <b>rkfixed</b> with <b>Adams</b>, <b>BDF</b>, or <b>AdamsBDF</b> and compare the results.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">In addition, the new solvers have the following optional arguments:</p>
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					<b>tol</b> - A real value or a vector of real values specifying tolerances for each independent variable in the system. You can use <b>tol</b> to override the default tolerance of 10<sup>-5</sup>.</p>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<b>J </b>(<b>BDF</b> and <b>AdamsBDF</b> only) - a function that returns the n <f family="Arial" charset="0" size="10">x</f> n Jacobian matrix, the matrix of partial derivatives of the system of equations in <b>D</b> with respect to the variables y<sub>0</sub>
					<b>, </b>y<sub>1</sub>
					<b>,</b>
					<b> ..., </b>y<sub>n</sub>. Supplying <b>J</b> can improve the accuracy of your results. </p>
			</text>
			<indexes>
				<phrase>J</phrase>
				<phrase>Jacobian matrix</phrase>
			</indexes>
		</region>
		<region region-id="805" left="36" top="756.75" width="237.75" height="14.25" align-x="36" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The <b>
						<u>
							<link href="#example" title="Links to example" popup="false">example</link>
						</u>
					</b> below uses J. </p>
			</text>
		</region>
		<region region-id="806" left="36" top="780.75" width="324.75" height="28.5" align-x="36" align-y="792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<b>
						<c val="#008080">Note</c>
					</b>
					<inlineAttr font-weight="normal">:</inlineAttr>
					<sp/>
					<b>J</b> is different from the matrix used as the final argument for the solvers <b>Stiffb</b> and <b>Stiffr</b>. </p>
			</text>
		</region>
		<region region-id="714" left="6" top="830.25" width="115.5" height="12" align-x="6" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="odesolve">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Changes to Odesolve</p>
			</text>
			<indexes>
				<phrase>Odesolve, changes</phrase>
				<phrase>Adams/BDF</phrase>
				<phrase>Radau</phrase>
			</indexes>
		</region>
		<region region-id="783" left="6" top="846.75" width="340.5" height="57" align-x="6" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">When you use the function <b>Odesolve</b> with a solve block to solve an ODE, right-clicking on <b>Odesolve</b> displays a drop-down menu from which you can select the solver you want to use. This menu has the following changes in Mathcad 14:</p>
			</text>
		</region>
		<region region-id="786" left="6" top="918.75" width="344.25" height="85.5" align-x="18.75" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<b>
						<c val="#008080">•</c>
					</b>
					<b>
						<c val="#000">
							<sp/>
							<tab/>
						</c>
					</b>The default solver is changed from <b>Fixed</b> to <b>Adams/BDF</b>, which calls the new and superior solver <b>
						<u>
							<link href="#AdamsBDF" title="Links to section" popup="false">AdamsBDF</link>
						</u>
					</b>. As a result, <b>Odesolve</b> may return different, but more accurate, results than in earlier versions of Mathcad.</p>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5"/>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<b>
						<c val="#008080">•</c>
					</b>
					<b>
						<c val="#000">
							<sp/>
							<tab/>
						</c>
					</b>The option <b>Stiff</b> is renamed <b>Radau</b>. </p>
			</text>
		</region>
		<region region-id="731" left="6" top="1014.75" width="346.5" height="85.5" align-x="9.75" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">Note</c>
					</b>: If you created a worksheet in Mathcad version 13 or earlier, in which you called <b>Odesolve</b> with <b>Fixed</b> as the selected solver, when you open the worksheet in Mathcad 14, Mathcad automatically changes the solver to <b>Adams/BDF</b>. However, if you create a new 14 worksheet using <b>Odesolve </b>
					<inlineAttr font-weight="normal">with</inlineAttr>
					<sp/>
					<b>Fixed</b> as the selected solver, Mathcad retains that setting when you re-open the worksheet.</p>
			</text>
		</region>
		<region region-id="774" left="0" top="1104" width="6000" height="6" align-x="0" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="891" left="6" top="1118.25" width="132" height="12" align-x="6" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="Radau">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Enhancements to Radau</p>
			</text>
			<indexes>
				<phrase>Radau, enhancements</phrase>
				<phrase>Radau, optional arguments</phrase>
			</indexes>
		</region>
		<region region-id="796" left="6" top="1134.75" width="342.75" height="28.5" align-x="6" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The solver <b>Radau</b>, which is designed to solve stiff systems, now accepts three new optional arguments:</p>
			</text>
		</region>
		<region region-id="794" left="6" top="1182.75" width="329.25" height="57" align-x="21.75" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<b>
						<c val="#008080">•</c>
					</b>
					<b>
						<c val="#000">
							<sp/>
							<tab/>
						</c>
					</b>
					<b>J -</b>
					<inlineAttr font-weight="normal"> A function that returns the n </inlineAttr>
					<f family="Arial" charset="0" size="10">
						<inlineAttr font-weight="normal">x</inlineAttr>
					</f>
					<inlineAttr font-weight="normal">
						<sp/>
					</inlineAttr>
					<inlineAttr font-weight="normal">n </inlineAttr>
					<inlineAttr font-weight="normal">Jacobian matrix, the matrix of partial derivatives of the system of equations in </inlineAttr>
					<b>D</b>.</p>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<b>
						<c val="#008080">•</c>
					</b>
					<b>
						<c val="#000">
							<sp/>
							<tab/>
						</c>
					</b>
					<b>M</b> - A real matrix representing a coupling between the variables of the form</p>
			</text>
		</region>
		<region region-id="809" left="36" top="1257" width="101.25" height="36.75" align-x="93.75" align-y="1278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:id xml:space="preserve">M</ml:id>
						<ml:parens>
							<ml:apply>
								<ml:derivative/>
								<ml:lambda>
									<ml:boundVars>
										<ml:id xml:space="preserve">t1</ml:id>
									</ml:boundVars>
									<ml:id xml:space="preserve">y</ml:id>
								</ml:lambda>
							</ml:apply>
						</ml:parens>
					</ml:apply>
					<ml:apply>
						<ml:id xml:space="preserve">f</ml:id>
						<ml:sequence>
							<ml:id xml:space="preserve">t1</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:sequence>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="4"/>
		</region>
		<region region-id="586" left="6" top="1308.75" width="341.25" height="45.75" align-x="11.25" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<b>
						<c val="#008080">•</c>
					</b>
					<b>
						<c val="#000">
							<sp/>
							<tab/>
						</c>
					</b>
					<b>tol</b> - A real value or a vector of real values specifying tolerances for each independent variable in the system. You can use <b>tol</b> to override the default tolerance of 10<sup>-5</sup>.</p>
			</text>
		</region>
		<region region-id="722" left="6" top="1368.75" width="293.25" height="42.75" align-x="14.25" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<b>Radau </b>
					<inlineAttr font-weight="normal">accepts these arguments using the syntax</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<tab/>
					<b>Radau</b>
					<b>(init, t1, t2, npoints, D, [J], [M], [tol])</b>
					<sp/>
				</p>
			</text>
		</region>
		<region region-id="737" left="6" top="1428.75" width="354" height="28.5" align-x="14.25" align-y="1440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The arguments <b>J </b>and <b>tol</b> are also available to the new solvers described in <b>
						<u>
							<link href="#new_solvers" title="Links to section" popup="false">New Solvers for Ordinary Differential Equations</link>
						</u>
					</b>. </p>
			</text>
		</region>
		<region region-id="811" left="6" top="1478.25" width="47.25" height="12" align-x="6" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="example">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Example</p>
			</text>
		</region>
		<region region-id="605" left="6" top="1494.75" width="262.5" height="14.25" align-x="26.25" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Suppose you want to solve a system of the following form:</p>
			</text>
		</region>
		<region region-id="659" left="36" top="1521" width="139.5" height="36.75" align-x="71.25" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:derivative/>
						<ml:lambda>
							<ml:boundVars>
								<ml:id xml:space="preserve">t1</ml:id>
							</ml:boundVars>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>0</ml:real>
							</ml:apply>
						</ml:lambda>
					</ml:apply>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:real>-.04</ml:real>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>0</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:apply>
									<ml:pow/>
									<ml:real>10</ml:real>
									<ml:real>4</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="5"/>
		</region>
		<region region-id="861" left="36" top="1563" width="201.75" height="36.75" align-x="71.25" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:derivative/>
						<ml:lambda>
							<ml:boundVars>
								<ml:id xml:space="preserve">t1</ml:id>
							</ml:boundVars>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>1</ml:real>
							</ml:apply>
						</ml:lambda>
					</ml:apply>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:real>.04</ml:real>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>0</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:apply>
									<ml:mult style="default"/>
									<ml:apply>
										<ml:pow/>
										<ml:real>10</ml:real>
										<ml:real>4</ml:real>
									</ml:apply>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:real>3</ml:real>
								<ml:apply>
									<ml:pow/>
									<ml:real>10</ml:real>
									<ml:real>7</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="6"/>
		</region>
		<region region-id="860" left="36" top="1608.75" width="99.75" height="17.25" align-x="49.5" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:real>0</ml:real>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:plus/>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>0</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:real>1</ml:real>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="7"/>
		</region>
		<region region-id="775" left="0" top="1632" width="6000" height="6" align-x="0" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="686" left="6" top="1638.75" width="342" height="28.5" align-x="6" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that while the first two equations involve derivatives, the third is algebraic. You can write this system in the following form: </p>
			</text>
		</region>
		<region region-id="608" left="42" top="1689" width="101.25" height="36.75" align-x="99.75" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:mult style="default"/>
						<ml:id xml:space="preserve">M</ml:id>
						<ml:parens>
							<ml:apply>
								<ml:derivative/>
								<ml:lambda>
									<ml:boundVars>
										<ml:id xml:space="preserve">t1</ml:id>
									</ml:boundVars>
									<ml:id xml:space="preserve">y</ml:id>
								</ml:lambda>
							</ml:apply>
						</ml:parens>
					</ml:apply>
					<ml:apply>
						<ml:id xml:space="preserve">f</ml:id>
						<ml:sequence>
							<ml:id xml:space="preserve">t1</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:sequence>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="8"/>
		</region>
		<region region-id="613" left="6" top="1740.75" width="151.5" height="14.25" align-x="19.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where f(t,y) is the vector function </p>
			</text>
		</region>
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								<ml:apply>
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									<ml:apply>
										<ml:mult style="default"/>
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											<ml:pow/>
											<ml:real>10</ml:real>
											<ml:real>4</ml:real>
										</ml:apply>
										<ml:apply>
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									<ml:apply>
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							<ml:apply>
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								<ml:apply>
									<ml:mult style="default"/>
									<ml:real>3</ml:real>
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										<ml:pow/>
										<ml:real>10</ml:real>
										<ml:real>7</ml:real>
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								</ml:apply>
								<ml:apply>
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								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
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								<ml:apply>
									<ml:plus/>
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										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>0</ml:real>
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									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:real>1</ml:real>
						</ml:apply>
					</ml:matrix>
				</ml:define>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and M is the "coupling matrix":</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">M</ml:id>
					<ml:matrix rows="3" cols="3">
						<ml:real>1</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
						<ml:real>1</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
						<ml:real>0</ml:real>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that M has a 0 in its lower-right corner because the left-hand side of the third equation is 0. As a result, </p>
			</text>
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								</ml:apply>
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						<ml:apply>
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								<ml:boundVars>
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								</ml:boundVars>
								<ml:apply>
									<ml:indexer/>
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									<ml:real>1</ml:real>
								</ml:apply>
							</ml:lambda>
						</ml:apply>
						<ml:real>0</ml:real>
					</ml:matrix>
				</ml:apply>
			</math>
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			<pageBreak/>
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		<region region-id="632" left="6" top="2118.75" width="341.25" height="42.75" align-x="12" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To improve the accuracy of your results, you can supply the Jacobian matrix <b>J</b> as an optional argument to <b>Radau</b>
					<inlineAttr font-weight="normal">.</inlineAttr> The Jacobian<inlineAttr font-weight="normal"> has the following form.</inlineAttr>
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								<ml:apply>
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								</ml:apply>
							</ml:lambda>
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										<ml:real>0</ml:real>
									</ml:apply>
								</ml:boundVars>
								<ml:apply>
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								</ml:apply>
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								<ml:apply>
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									<ml:real>2</ml:real>
								</ml:apply>
							</ml:lambda>
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						<ml:apply>
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								<ml:apply>
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								<ml:apply>
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						<ml:apply>
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									<ml:apply>
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										<ml:real>1</ml:real>
									</ml:apply>
								</ml:boundVars>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">f</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:lambda>
						</ml:apply>
						<ml:apply>
							<ml:derivative/>
							<ml:lambda>
								<ml:boundVars>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:boundVars>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">f</ml:id>
									<ml:real>0</ml:real>
								</ml:apply>
							</ml:lambda>
						</ml:apply>
						<ml:apply>
							<ml:derivative/>
							<ml:lambda>
								<ml:boundVars>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:boundVars>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">f</ml:id>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:lambda>
						</ml:apply>
						<ml:apply>
							<ml:derivative/>
							<ml:lambda>
								<ml:boundVars>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">y</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:boundVars>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">f</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:lambda>
						</ml:apply>
					</ml:matrix>
				</ml:apply>
			</math>
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			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In this example, the Jacobian is</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:function>
						<ml:id xml:space="preserve">J</ml:id>
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							<ml:id xml:space="preserve">y</ml:id>
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						<ml:real>-.04</ml:real>
						<ml:real>.04</ml:real>
						<ml:real>1</ml:real>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:pow/>
								<ml:real>10</ml:real>
								<ml:real>4</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
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							<ml:apply>
								<ml:mult style="default"/>
								<ml:apply>
									<ml:neg/>
									<ml:apply>
										<ml:pow/>
										<ml:real>10</ml:real>
										<ml:real>4</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:apply>
									<ml:mult style="default"/>
									<ml:real>6</ml:real>
									<ml:apply>
										<ml:pow/>
										<ml:real>10</ml:real>
										<ml:real>7</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:indexer/>
									<ml:id xml:space="preserve">y</ml:id>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:real>1</ml:real>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:pow/>
								<ml:real>10</ml:real>
								<ml:real>4</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>1</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult style="default"/>
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:pow/>
									<ml:real>10</ml:real>
									<ml:real>4</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:indexer/>
								<ml:id xml:space="preserve">y</ml:id>
								<ml:real>1</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:real>1</ml:real>
					</ml:matrix>
				</ml:define>
			</math>
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		</region>
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			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Create a vector yinit of initial conditions for y<sub>0</sub>, y<sub>1</sub>, and y<sub>2</sub>.</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">yinit</ml:id>
						<ml:real>0</ml:real>
					</ml:apply>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="14"/>
		</region>
		<region region-id="642" left="30" top="2490.75" width="48" height="17.25" align-x="61.5" align-y="2502" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">yinit</ml:id>
						<ml:real>1</ml:real>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="15"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">yinit</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
					<ml:real>0</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="16"/>
		</region>
		<region region-id="653" left="6" top="2556.75" width="312" height="14.25" align-x="21" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Create endpoints for the interval over which the ODE is to be solved.</p>
			</text>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now, apply the solver Radau with these arguments.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The following plots show the graph of the solution.</p>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">New State Space Solver</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad 14 features a new state space solver, <b>statespace</b>, that solves systems of the form:<sp count="2"/>
				</p>
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			<indexes>
				<phrase>statespace</phrase>
				<phrase>state space solver</phrase>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">where</p>
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					<b>
						<c val="#008080">•</c>
					</b>
					<sp count="2"/>x(t) is a vector of unknown states.</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>A(t) is the state matrix.</p>
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					<b>
						<c val="#008080">•</c>
					</b>
					<sp count="2"/>B(t) is the input matrix.</p>
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					<b>
						<c val="#008080">•</c>
					</b>
					<sp count="2"/>u(t) is the input, or control, function. u(t) can be a scalar or vector-valued function.<sp count="3"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The following example illustrates the <b>statespace</b> solver.</p>
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					<b>statespace</b> takes<sp count="2"/>the following arguments:</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
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					<tab/>statespace(x0, t0, t1, npts, A, B, u)</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>x0 is a vector of initial conditions.</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>t0 is a starting point of the integration interval.</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>t1 is an ending point of the integration interval.</p>
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					<b>
						<c val="#008080">•</c>
					</b>
					<sp count="2"/>npts is the number of points at which to return results.</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>A is the state matrix.</p>
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					<b>
						<c val="#008080">•</c>
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					<sp count="2"/>B is the input matrix.</p>
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					<sp count="2"/>u is the control function. u can be a scalar or a vector.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The graph of the solution is shown below.</p>
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