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      <text>25.3.1.3 Solution of important PDEs</text>
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      <text><![CDATA[<p>Consider now a similar type of PDE i.e. one that can also be solved by direct integration.
</p>
Suppose we require the general solution of
<p> <span class="math inline">\(\qquad\displaystyle \dfrac{\partial^2u}{\partial
  x^2}=2xe^t\)</span></p> where <span class="math inline">\(u\)</span> is a function of <span class="math inline">\(x\)</span> and <span class="math inline">\(t\)</span>.
<p></p>
<p>Integrating with respect to <span class="math inline">\(x\)</span> gives us </p>
<p><span class="math inline">\(\qquad\displaystyle \dfrac{\partial
  u}{\partial x}=x^2e^t+f(t)\)</span></p> where the arbitrary function <span class="math inline">\(f(t)\)</span> replaces the normal “arbitrary constant” of ordinary integration. This function of <span class="math inline">\(t\)</span> only is needed because
we are integrating “partially” with respect to <span class="math inline">\(x\)</span> i.e. we are reversing a partial differentiation with respect to <span class="math inline">\(x\)</span> at constant <span class="math inline">\(t\)</span>.
<p></p>
<p>Integrating again with respect to <span class="math inline">\(x\)</span> gives the general solution: </p>
<p><span class="math inline">\(\qquad\displaystyle u=\dfrac{x^3}{3}e^t+x\
  f(t)+g(t)\)</span></p> where <span class="math inline">\(g(t)\)</span> is a second arbitrary function. We have now obtained the general solution of the given PDE but to find the arbitrary function we must know two initial conditions.
<p></p>
<p>Suppose, for the sake of example, that these conditions are <span class="math inline">\(\qquad\displaystyle  u (0,t)=t\ ,\qquad
\dfrac{\partial u}{\partial x}(0,t)=e^t\)</span></p>
<p> Inserting the first of these conditions into the general solution gives<span class="math inline">\(\ \ \displaystyle g(t)=t\)</span>.</p>
<p>Inserting the second condition into the general solution gives
    <span class="math inline">\(\ \ \displaystyle f(t)=e^t\)</span>.</p> So the final solution is<span class="math inline">\(\ \ \displaystyle
u=\dfrac{x^3}{3}e^t+xe^t+t.\)</span>]]></text>
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