Section 14.3 Partial Derivatives

Show that each of the following functions is a solution of the wave equation
\[u_{tt} = a^2 u_{xx}.\]

1. \(u = \sin(kx)\sin(akt)\)
   
2. \(u = \frac{t}{a^2 t^2 - x^2}\)
   
3. \(u = (x - at)^6 + (x + at)^6\)
   
4. \(u = \sin(x - at) + \ln(x + at)\)