I am working on a mock exam for Partial differential equations, and the exam claims that the following two functions
$p(z,t) = \frac{1}{2\lambda} e^{-\lambda z + \lambda^2 t / 2}$
and
$p(z,t) = \frac{1}{\sqrt{2\pi t}} e^{-z^2/(2t)}$
satisfy the equation
$\frac{\partial p}{\partial t} - \frac{1}{2}\frac{\partial^2 p}{\partial z^2}=0 ?$
But I don't think they do.
For the first one, my "solution" was to note that:
$\frac{\partial p}{\partial t} = -\frac{\lambda^2}{2}p(z,t)$ and $\frac{\partial^2 p}{\partial z^2} = \lambda^2p(z,t)$
Hence, the equation does not hold.
What am I missing?