please help me in solving this 2nd order differential equations...thanks in advance
Well, since they have been so nice and given you the solution I guess you can assume that it will be on the form $\displaystyle u=e^{Ax+Bt}$, where A and B are constants.
$\displaystyle \frac{du}{dt}=Be^{Ax+Bt}$
$\displaystyle \frac{du}{dx}=Ae^{Ax+Bt}$
$\displaystyle \frac{d^2u}{dx^2}=A^2e^{Ax+Bt}$
From here it is just put substitute the terms in the equation that is given, with the right hand sides here, and solve the equation
Unfortunately, you have two unknown but only one equation, so you are one equation short. I guess something is missing in the problem statement.