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@@ -140,19 +145,19 @@ Functions</h2></td></tr>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Solve a multivariable first order <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation" target="_blank">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Euler_method" target="_blank">forward Euler method</a> </p>
<dl class="section author"><dt>Authors</dt><dd><a href="https://github.com/kvedala" target="_blank">Krishna Vedala</a> </dd></dl>
<p>The ODE being solved is: </p><p class="formulaDsp">
<p>The ODE being solved is: </p><p class="formulaDsp">
\begin{eqnarray*}
\dot{u} &amp;=&amp; v\\
\dot{v} &amp;=&amp; -\omega^2 u\\
\omega &amp;=&amp; 1\\
[x_0, u_0, v_0] &amp;=&amp; [0,1,0]\qquad\ldots\text{(initial values)}
\end{eqnarray*}
\dot{u} &amp;=&amp; v\\
\dot{v} &amp;=&amp; -\omega^2 u\\
\omega &amp;=&amp; 1\\
[x_0, u_0, v_0] &amp;=&amp; [0,1,0]\qquad\ldots\text{(initial values)}
\end{eqnarray*}
</p>
<p> The exact solution for the above problem is: </p><p class="formulaDsp">
<p> The exact solution for the above problem is: </p><p class="formulaDsp">
\begin{eqnarray*}
u(x) &amp;=&amp; \cos(x)\\
v(x) &amp;=&amp; -\sin(x)\\
\end{eqnarray*}
u(x) &amp;=&amp; \cos(x)\\
v(x) &amp;=&amp; -\sin(x)\\
\end{eqnarray*}
</p>
<p> The computation results are stored to a text file <code>forward_euler.csv</code> and the exact soltuion results in <code>exact.csv</code> for comparison. <img src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/ode_forward_euler.svg" alt="Implementation solution" style="pointer-events: none;" class="inline"/></p>
<p>To implement <a href="https://en.wikipedia.org/wiki/Van_der_Pol_oscillator" target="_blank">Van der Pol oscillator</a>, change the <a class="el" href="#abaeae8f62a018d197f0187a1c80a90fe" title="Problem statement for a system with first-order differential equations. Updates the system differenti...">problem</a> function to: </p><div class="fragment"><div class="line"><span class="keyword">const</span> <span class="keywordtype">double</span> mu = 2.0;</div>
@@ -169,7 +174,7 @@ v(x) &amp;=&amp; -\sin(x)\\
<tr>
<td class="memname">void exact_solution </td>
<td>(</td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>x</em>, </span></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>x</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>argc</em>, </span></td>
<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>argc</em></span>, </td>
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<td class="paramkey"></td>
<td></td>
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<td class="memname">void problem </td>
<td>(</td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>x</em>, </span></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>x</em></span>, </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *</td> <td class="paramname"><span class="paramname"><em>y</em>, </span></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *</td> <td class="paramname"><span class="paramname"><em>y</em></span>, </td>
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<td class="paramkey"></td>
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<tr>
<td class="memname">void save_exact_solution </td>
<td>(</td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>X0</em>, </span></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>X0</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>X_MAX</em>, </span></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>X_MAX</em></span>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>step_size</em>, </span></td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>step_size</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
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