First solve to get the homogenous solution. Can you do this?

You're given that the particular solution has the form (you should think about why that is) so you need to get the value of A. Start by calculating and . Can you do this?

Note: Once you have the value of A and hence the particular solution you can construct the complete solution by adding together the homogenous and particular solutions. The boundary conditions x = 1 and at t = 0 are used to get the value of the arbitrary constants.