I looked at this for a while and cant get a complete solution, but since no one replied for 4 days ill point out that if you divide your FOC for c(t) by the FOC for c(t-1) you get the relationship between c(t) and c(t-1).

Then you need to choose the value of c(1) that makes the whole progression fit within the budget constraint, although i couldn't work out how...so i may be pointing you in the wrong direction.