Solución

Primero, calculamos $u_{tt}, u_{xx}$ y $u_{yy}$ :

\begin{aligned} u_{tt} &= \frac{\partial }{\partial t}\bigg[\frac{\partial u}{\partial t} \bigg]\\ &= \frac{\partial }{\partial t}[5sen(3\pi x)sen(4\pi y)(−10\pi sen(10\pi t))]\\ &= \frac{\partial }{\partial t}[−50\pi sen(3\pi x)sen(4\pi y)sen(10\pi t)]\\ &= −500\pi^2 sen(3\pi x)sen(4\pi y)cos(10\pi t) \end{aligned}

\begin{aligned} u_{xx} &= \frac{\partial }{\partial x}\bigg[\frac{\partial u}{\partial x} \bigg]\\ &= \frac{\partial }{\partial x}[15\pi cos(3\pi x)sen(4\pi y)cos(10\pi t)]\\ &= −45\pi^2sen(3\pi x)sen(4\pi y)cos(10\pi t) \end{aligned}

\begin{aligned} u_{yy} &= \frac{\partial }{\partial y}\bigg[\frac{\partial u}{\partial y} \bigg]\\ &= \frac{\partial }{\partial y} [5sen(3\pi x)(4\pi cos(4\pi y))cos(10\pi t)]\\ &= \frac{\partial }{\partial y} [20\pi sen(3\pi x)cos(4\pi y)cos(10\pi t)]\\ &= −80\pi^2sen(3\pi x)sen(4\pi y)cos(10\pi t) \end{aligned}

A continuación, sustituimos cada uno de estos en el lado derecho de la Ecuación 4.20 y simplificamos:

4(u_{xx}+u_{yy}) = 4(−45\pi^2 sen(3\pi x)sen(4\pi y)cos(10\pi t)

−80\pi^2sen(3\pi x)sen(4\pi y)cos(10\pi t))

= 4(−125\pi^2 sen(3\pi x)sen(4\pi y)cos(10\pi t))

= −500\pi^2sen(3\pi x)sen(4\pi y)cos(10\pi t)= u_{tt}

Esto verifica la solución.