Please how can I verify that $$u(x,t)=\frac{1}{\sqrt{2\pi Dt}}\exp\left(-\frac{(x-x_{0}-ct)^{2}}{2Dt} \right) $$ is a solution of the diffusion equation with drift: $$\frac{\partial u}{\partial t}=-c\frac{\partial u}{\partial x}+\frac{D}{2}\frac{\partial^{2} u}{\partial x^{2}}$$ for $t\in (0,\infty)$ and $x\in (-\infty,\infty)$
The probability $u(x,t)=Prob\{X(t)=x\}$ represents the p.d.f. of a continuous-time and continuous-state process $X(t)$,
When you have a specific function that could be a solution to a given differential equation, the simple and straight forward way to verify if it is indeed a solution would be to substitute it in the expression and carry out the calculations to see if the equation is true.
In this particular case, you carry out the following differentiations first
$$\dfrac{\partial u}{\partial t}, \dfrac{\partial u}{\partial x} \quad \text{and} \quad \dfrac{\partial^2 u}{\partial^2x}$$
so that you don't get a big complicated phrase and then check if the equation holds. Specifically, where $u$ in the given differential equation, you will substitute $u(x,t)$. Can you proceed?