Given: $$ z= A \exp(pt) \sin(px) $$ where $A,p$ are arbitrary constants to be eliminated. Find the associated PDE.
If I am not wrong since there are 2 constants and 2 independent variables we must get a 1st order PDE which is unique right?
My book gives the answer in the form of a second order PDE.
Note that taking a $t$-derivative adds a factor of $p$, and $x$-derivative also flips between sine and cosine, so the relation we are seeking will deal with 2 derivatives, since that will flip the sine twice back into the sine. We get $$ z_{tt} = p^2z \quad \text{and} \quad z_{xx} = -p^2z, $$ so the PDE you seek is $$z_{tt} + z_{xx} = 0.$$