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      <text>25.3.2.4  Method of separation of variables - general approach</text>
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      <text><![CDATA[<b>Laplace’s equation</b>

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            <td>\(\qquad\displaystyle \dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0\)
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            <td>\((9)\)</td>
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<h4 class="HELM_example">Example</h4>
<p>Separating the variables for Laplace’s equation follows similar lines to the previous Example. Obtain the ODEs satisfied by <span class="math inline">\(X(x)\)</span> and <span class="math inline">\(Y(y)\)</span>.</p>
<h4 class="HELM_solution">Solution</h4>
<p>Assuming \(u(x, y) = X(x)Y(y)\) leads to: \(\displaystyle \dfrac{\partial^2u}{\partial x^2}=X''Y\) \(\qquad\displaystyle \dfrac{\partial^2u}{\partial y^2}=XY''\) so
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<p>\(\qquad X''Y + XY''=0\) \(\qquad\) or \(\qquad\) \(\dfrac{X''}{X}=-\dfrac{Y''}{Y}\)
    <br>Equating each side to a constant \(K\)
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            <td>\(\qquad \dfrac{X''}{X}=K\)\(\qquad\) or\(\qquad\) \(\displaystyle \frac{\mathrm{d^2}X}{\mathrm{d}x^2}-KX=0\)</td>
            <td>\((10a)\)</td>
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            <td>\(\qquad \dfrac{Y''}{Y}=-K\)\(\qquad\) or\(\qquad\)\(\displaystyle \frac{\mathrm{d^2}Y}{\mathrm{d}y^2}+KY=0\)</td>
            <td>\((10b)\)</td>
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<p>(Note carefully the different signs in the two ODEs. Yet again the sign of the “separation constant” \(K\) will determine the solutions.)
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