I take it that you mean to find a and b such that $\displaystyle u(x,t)=ex+at(6x+bt)$ is a solution to $\displaystyle \frac{\partial^2u}{\partial t^2}=172\frac{\partial^2u}{\partial x^2}$?

well, find $\displaystyle \frac{\partial^2u}{\partial t^2}$ and $\displaystyle \frac{\partial^2u}{\partial t^2}$.

$\displaystyle u(x,t)=ex+6axt+abt^2$

$\displaystyle \frac{\partial u}{\partial t}=6ax+2abt$

$\displaystyle \frac{\partial^2 u}{\partial t^2}=2ab$

$\displaystyle \frac{\partial u}{\partial x}=e+6at$

$\displaystyle \frac{\partial^2 u}{\partial x^2}=0$

Thus, $\displaystyle 2ab=0$...

...are you sure you wrote out the problem right?

I'm getting zero as an answer for both!

--Chris