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      <text>25.4.1.3</text>
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      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Now use the separation of variables method, putting \(u(x,y)=X(x)Y(y)\), to find the differential equations satisfied by \(X(x)\), \(Y(y)\) and decide on the sign of the separation constant \(K\):</p>
<h4 class="HELM_solution">Solution</h4>
<p>We have boundary conditions which, like the worked example above, are periodic in \(x\). Hence the differential equations are, again,

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            <td>\(\qquad X''=-\lambda^2 X\)</td>
            <td>\(Y''=+\lambda^2\)</td>
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putting the separation constant \(K\) as \(-\lambda^2\).
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