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      <text>25.3.2.1  Method of separation of variables - general approach</text>
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      <text><![CDATA[<h3>Method of separation of variables - general approach</h3>

<p>In Section 25.2 we showed that</p>
<ol type="a">

    <ol type="a">
        <li>
            <p><span class="math inline">\(u(x,y)=\sin x \cosh y\)</span> <br>is
                <span><strong>a</strong></span> solution of the two-dimensional Laplace equation
            </p>
        </li>
        <li>
            <p><span class="math inline">\(u(x,t)=e^{-2\pi^2t}\sin \pi
x\)</span> <br>is <span><strong>a</strong></span> solution of the one-dimensional heat conduction equation</p>
        </li>
        <li>
            <p><span class="math inline">\(u(x,t)=u_0\sin \left(\dfrac{\pi
x}{\ell}\right)\cos \left(\dfrac{\pi c
t}{\ell}\right)\)</span> <br>is <span><strong>a</strong></span> solution of the one-dimensional wave equation.</p>
        </li>
    </ol>

</ol>
<p>All three solutions here have a specific form: in (a) <span class="math inline">\(u(x,y)\)</span> is a product of a function of
    <span class="math inline">\(x\)</span> alone, <span class="math inline">\(\sin x\)</span>, and a function of <span class="math inline">\(y\)</span> alone, <span class="math inline">\(\cosh y\)</span>. Similarly, in both (b) and (c),
    <span class="math inline">\(u(x,t)\)</span> is a product of a function of <span class="math inline">\(x\)</span> alone and a function of <span class="math inline">\(t\)</span> alone. </p><br>The method of separation of variables involves finding solutions
of PDEs which are of this product form. In the method we assume that a solution to a PDE has the form.
<p></p>

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            <td>\(\qquad\displaystyle u(x,t)=X(x) T(t)\) </td>
            <td>\(\text{or}\qquad u(x,y)=X(x) Y(y))\)</td>
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    </tbody>
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<p>where <span class="math inline">\(X(x)\)</span> is a function of <span class="math inline">\(x\)</span> only, <span class="math inline">\(T(t)\)</span> is a function of <span class="math inline">\(t\)</span> only and <span class="math inline">\(Y(y)\)</span>    is a function <span class="math inline">\(y\)</span> only. You should note that not all solutions to PDEs are of this type; for example, it is easy to verify that </p>
<p><span class="math inline">\(\qquad\displaystyle
  u(x,y)=x^2-y^2\)</span> </p>(which is <span><strong>not</strong></span> of the form <span class="math inline">\(u(x,y)=X(x) Y(y)\)</span>) is a solution of the Laplace equation. <br>However, many interesting and useful solutions of PDEs are obtainable
which are of the product form. We shall firstly consider the types of solution obtainable for our three basic PDEs using trial solutions of the product form.]]></text>
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