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      <text>25.4.1.5</text>
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      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Impose the derivative boundary conditions on this solution:</p>
<h4 class="HELM_solution">Solution</h4>
<p></p>
<p>\(\qquad \dfrac{\partial u}{\partial x}=(-\lambda A \sin \lambda x +\lambda B\cos \lambda x)(Ce^{\lambda y}+D e^{-\lambda y})\)</p> <br> Hence\(\dfrac{\partial u}{\partial x}(0, y) = 0\) gives \(\lambda B(Ce^{\lambda y}+D e^{-\lambda y}) = 0\) for all
\(y.\)
<br>The possibility \(\lambda = 0\) can be excluded this would give a trivial constant solution in \((4)\). Hence we must choose \(B = 0\). <br> The condition \(\dfrac{\partial u}{\partial x}(1, y) = 0\) gives
<p> \(\qquad -\lambda A\sin\lambda(Ce^{\lambda y}+D e^{-\lambda y}) = 0\)</p> Choosing \(A = 0\) would make \(u \equiv 0\) so we must force \(\sin \lambda\) to be zero i.e. choose \(\lambda = nπ\) where \(n\) is a positive integer.<br> Thus, at this stage \((4)\)
becomes

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            <td>\(\qquad u = A \cos nπx(Ce^{nπy} + De^{-nπy})\)</td>
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            <td>\(&nbsp; &nbsp; &nbsp;\qquad = \cos nπx(Ee^{nπy} + F e^{−nπy})\)</td>
            <td>\[(5)\]</td>
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</table>]]></text>
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