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      <text>25.2.5 Other important PDEs in science and engineering</text>
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      <text><![CDATA[<h3>Other important PDEs in science and engineering</h3>
<ol>
    <li><strong>Poisson’s equation</strong>
        <p>\(\qquad \dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=f(x,y)\) \(\qquad\) (two-dimensional form)</p> where \(f(x,y)\) is a given function. This equation arises in electrostatics, elasticity theory and elsewhere.</li>
    <li><strong>Helmholtz’s equation</strong>
        <p>\(\qquad \dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}+k^2u=0\) \(\qquad\) (two dimensional form)</p> which arises in wave theory.</li>
    <li><strong>Schrödinger’s equation</strong>
        <p>\(\qquad -\dfrac{h^2}{8\pi^2 m}\left(\dfrac{\partial^2\psi}{\partial x^2}+\dfrac{\partial^2\psi}{\partial y^2}+\dfrac{\partial^2\psi}{\partial z^2}\right)=E\psi\)</p> which arises in quantum mechanics. (\(h\) is Planck’s constant)
    </li>
    <li><strong>Transverse vibrations equation</strong>
        <p>\(\qquad a^2\dfrac{\partial^4 u}{\partial x^4}+\dfrac{\partial^2u}{\partial t^2}=0\)</p> for a homogeneous rod, where \(u(x,t)\) is the displacement at time \(t\) of the cross section through \(x\).</li>
</ol>
<p>All the PDEs we have discussed are <strong>second order</strong> (because the highest order derivatives that arise are second order) apart from the last example which is <strong>fourth order</strong>.</p>
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