Chapter 1 - Introduction - 1.3 Classification of Differential Equations - Problems - Page 25: 27

$u_{1,x}=\lambda \cos\lambda x\sin\lambda at,\ u_{1,xx}=-\lambda^2 \sin\lambda x\sin\lambda at$ $u_{1,t}=a\lambda\sin\lambda x\cos\lambda at,\ u_{1,tt}=-a^2\lambda^2\sin\lambda x\sin\lambda at$ $a^2u_{1,xx}=u_{1,tt}$ $u_{2,x}=\cos(at-x),\ u_{2,xx}=\sin(at-x)$ $u_{2,t}=-a\cos(at-x),\ u_{2,tt}=a^2\sin(at-x)$ $a^2u_{2,xx}=u_{2,tt}$